MathWiki - User contributions [en]

https://math.byu.edu/wiki/api.php?action=feedcontributions&feedformat=atom&user=Jsc3 MathWiki - User contributions [en] 2026-07-18T09:49:00Z User contributions MediaWiki 1.26.3 https://math.byu.edu/wiki/index.php?title=Math_355:_Graph_Theory&diff=3653 Math 355: Graph Theory 2017-11-09T20:05:14Z

<p>Jsc3: /* Additional topics */</p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Graph Theory.<br /> <br /> === (Credit Hours:Lecture Hours:Lab Hours) ===<br /> (3:3:0)<br /> <br /> === Offered ===<br /> W<br /> <br /> === Prerequisite ===<br /> [[Math 313]].<br /> <br /> === Description ===<br /> Graphs and subgraphs, coloring problems, applications.<br /> <br /> == Desired Learning Outcomes ==<br /> <br /> === Prerequisites ===<br /> <br /> Math 313.<br /> <br /> === Minimal learning outcomes ===<br /> Definition of a graph, examples (trees, paths, cycles, Petersen graph, etc.), First Theorem of Graph Theory, isomorphisms of graphs and graph invariants, chromatic number, connectivity, planar graphs and conditions for planarity, statements of Four Color Theorem and Kuratowski's theorem, Hamiltonian cycles.<br /> <br /> Additional topics from 1) proof of Kuratowski's theorem, 2) definition of the Euler characteristic and proof of Euler's formula for the genus, 3) proof of the Chvatal-Erdos theorem on the existence of Hamiltonian cycles.<br /> <div style="-moz-column-count:2; column-count:2;"><br /> <br /> </div><br /> <br /> === Textbooks ===<br /> Possible textbooks for this course include (but are not limited to):<br /> <br /> * ''Graph Theory'' by Rusell Merris (Chap. 1-5).<br /> <br /> A good additional resource is An Introduction to Graph Theory by Douglas B. West, but the book is probably too encyclopedic to use as a main text.<br /> <br /> === Additional topics ===<br /> <br /> <br /> Possible other topics include spectral graph theory (networkings, expanders, Ramanujan graphs), characterization of Ramanujan graphs by the Riemann hypothesis for its zeta function. A good source for this topic is the paper by M. Ram Murty, Ramanujan Graphs, J. Ramanujan Math. Soc. 18(2003) 1-20.<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|355]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_355:_Graph_Theory&diff=3652 Math 355: Graph Theory 2017-11-09T20:04:40Z

<p>Jsc3: /* Additional topics */</p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Graph Theory.<br /> <br /> === (Credit Hours:Lecture Hours:Lab Hours) ===<br /> (3:3:0)<br /> <br /> === Offered ===<br /> W<br /> <br /> === Prerequisite ===<br /> [[Math 313]].<br /> <br /> === Description ===<br /> Graphs and subgraphs, coloring problems, applications.<br /> <br /> == Desired Learning Outcomes ==<br /> <br /> === Prerequisites ===<br /> <br /> Math 313.<br /> <br /> === Minimal learning outcomes ===<br /> Definition of a graph, examples (trees, paths, cycles, Petersen graph, etc.), First Theorem of Graph Theory, isomorphisms of graphs and graph invariants, chromatic number, connectivity, planar graphs and conditions for planarity, statements of Four Color Theorem and Kuratowski's theorem, Hamiltonian cycles.<br /> <br /> Additional topics from 1) proof of Kuratowski's theorem, 2) definition of the Euler characteristic and proof of Euler's formula for the genus, 3) proof of the Chvatal-Erdos theorem on the existence of Hamiltonian cycles.<br /> <div style="-moz-column-count:2; column-count:2;"><br /> <br /> </div><br /> <br /> === Textbooks ===<br /> Possible textbooks for this course include (but are not limited to):<br /> <br /> * ''Graph Theory'' by Rusell Merris (Chap. 1-5).<br /> <br /> A good additional resource is An Introduction to Graph Theory by Douglas B. West, but the book is probably too encyclopedic to use as a main text.<br /> <br /> === Additional topics ===<br /> <br /> <br /> Possible other topics include spectral graph theory (networkings, expanders, Ramanujan graphs), characterization of Ramanujan graphs by the Riemann hypothesis of its zeta function. A good source for this topic is the paper by M. Ram Murty, Ramanujan Graphs, J. Ramanujan Math. Soc. 18(2003) 1-20.<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|355]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_355:_Graph_Theory&diff=3651 Math 355: Graph Theory 2017-11-09T18:15:31Z

<p>Jsc3: /* Additional topics */</p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Graph Theory.<br /> <br /> === (Credit Hours:Lecture Hours:Lab Hours) ===<br /> (3:3:0)<br /> <br /> === Offered ===<br /> W<br /> <br /> === Prerequisite ===<br /> [[Math 313]].<br /> <br /> === Description ===<br /> Graphs and subgraphs, coloring problems, applications.<br /> <br /> == Desired Learning Outcomes ==<br /> <br /> === Prerequisites ===<br /> <br /> Math 313.<br /> <br /> === Minimal learning outcomes ===<br /> Definition of a graph, examples (trees, paths, cycles, Petersen graph, etc.), First Theorem of Graph Theory, isomorphisms of graphs and graph invariants, chromatic number, connectivity, planar graphs and conditions for planarity, statements of Four Color Theorem and Kuratowski's theorem, Hamiltonian cycles.<br /> <br /> Additional topics from 1) proof of Kuratowski's theorem, 2) definition of the Euler characteristic and proof of Euler's formula for the genus, 3) proof of the Chvatal-Erdos theorem on the existence of Hamiltonian cycles.<br /> <div style="-moz-column-count:2; column-count:2;"><br /> <br /> </div><br /> <br /> === Textbooks ===<br /> Possible textbooks for this course include (but are not limited to):<br /> <br /> * ''Graph Theory'' by Rusell Merris (Chap. 1-5).<br /> <br /> A good additional resource is An Introduction to Graph Theory by Douglas B. West, but the book is probably too encyclopedic to use as a main text.<br /> <br /> === Additional topics ===<br /> <br /> <br /> Possible other topics include spectral graph theory (networking, expanders, Ramanujan graphs), characterization of Ramanujan graphs by the Riemann hypothesis of its zeta function. A good source for this topic is the paper by M. Ram Murty, Ramanujan Graphs, J. Ramanujan Math. Soc. 18(2003) 1-20.<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|355]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_355:_Graph_Theory&diff=3650 Math 355: Graph Theory 2017-11-09T18:15:09Z

<p>Jsc3: /* Additional topics */</p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Graph Theory.<br /> <br /> === (Credit Hours:Lecture Hours:Lab Hours) ===<br /> (3:3:0)<br /> <br /> === Offered ===<br /> W<br /> <br /> === Prerequisite ===<br /> [[Math 313]].<br /> <br /> === Description ===<br /> Graphs and subgraphs, coloring problems, applications.<br /> <br /> == Desired Learning Outcomes ==<br /> <br /> === Prerequisites ===<br /> <br /> Math 313.<br /> <br /> === Minimal learning outcomes ===<br /> Definition of a graph, examples (trees, paths, cycles, Petersen graph, etc.), First Theorem of Graph Theory, isomorphisms of graphs and graph invariants, chromatic number, connectivity, planar graphs and conditions for planarity, statements of Four Color Theorem and Kuratowski's theorem, Hamiltonian cycles.<br /> <br /> Additional topics from 1) proof of Kuratowski's theorem, 2) definition of the Euler characteristic and proof of Euler's formula for the genus, 3) proof of the Chvatal-Erdos theorem on the existence of Hamiltonian cycles.<br /> <div style="-moz-column-count:2; column-count:2;"><br /> <br /> </div><br /> <br /> === Textbooks ===<br /> Possible textbooks for this course include (but are not limited to):<br /> <br /> * ''Graph Theory'' by Rusell Merris (Chap. 1-5).<br /> <br /> A good additional resource is An Introduction to Graph Theory by Douglas B. West, but the book is probably too encyclopedic to use as a main text.<br /> <br /> === Additional topics ===<br /> <br /> <br /> Possible other topics include spectral graph theory (networking, expanders, Ramanujan graphs), characterization of Ramanujan graphs by the Riemann hypothesis of its zeta function.A good source for this topic is the paper by M. Ram Murty, Ramanujan Graphs, J. Ramanujan Math. Soc. 18(2003) 1-20.<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|355]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_355:_Graph_Theory&diff=3649 Math 355: Graph Theory 2017-11-09T18:14:26Z

<p>Jsc3: /* Additional topics */</p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Graph Theory.<br /> <br /> === (Credit Hours:Lecture Hours:Lab Hours) ===<br /> (3:3:0)<br /> <br /> === Offered ===<br /> W<br /> <br /> === Prerequisite ===<br /> [[Math 313]].<br /> <br /> === Description ===<br /> Graphs and subgraphs, coloring problems, applications.<br /> <br /> == Desired Learning Outcomes ==<br /> <br /> === Prerequisites ===<br /> <br /> Math 313.<br /> <br /> === Minimal learning outcomes ===<br /> Definition of a graph, examples (trees, paths, cycles, Petersen graph, etc.), First Theorem of Graph Theory, isomorphisms of graphs and graph invariants, chromatic number, connectivity, planar graphs and conditions for planarity, statements of Four Color Theorem and Kuratowski's theorem, Hamiltonian cycles.<br /> <br /> Additional topics from 1) proof of Kuratowski's theorem, 2) definition of the Euler characteristic and proof of Euler's formula for the genus, 3) proof of the Chvatal-Erdos theorem on the existence of Hamiltonian cycles.<br /> <div style="-moz-column-count:2; column-count:2;"><br /> <br /> </div><br /> <br /> === Textbooks ===<br /> Possible textbooks for this course include (but are not limited to):<br /> <br /> * ''Graph Theory'' by Rusell Merris (Chap. 1-5).<br /> <br /> A good additional resource is An Introduction to Graph Theory by Douglas B. West, but the book is probably too encyclopedic to use as a main text.<br /> <br /> === Additional topics ===<br /> <br /> <br /> Possible other topics include spectral graph theory (networking, expanders, Ramanujan graphs), characterization of Ramanujan graphs by the Riemann hypothesis of its zeta function.<br /> A good source for this topic is the paper by M. Ram Murty, Ramanujan Graphs, J. Ramanujan Math. Soc. 18(2003) 1-20.<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|355]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_355:_Graph_Theory&diff=3648 Math 355: Graph Theory 2017-11-09T18:12:19Z

<p>Jsc3: /* Additional topics */</p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Graph Theory.<br /> <br /> === (Credit Hours:Lecture Hours:Lab Hours) ===<br /> (3:3:0)<br /> <br /> === Offered ===<br /> W<br /> <br /> === Prerequisite ===<br /> [[Math 313]].<br /> <br /> === Description ===<br /> Graphs and subgraphs, coloring problems, applications.<br /> <br /> == Desired Learning Outcomes ==<br /> <br /> === Prerequisites ===<br /> <br /> Math 313.<br /> <br /> === Minimal learning outcomes ===<br /> Definition of a graph, examples (trees, paths, cycles, Petersen graph, etc.), First Theorem of Graph Theory, isomorphisms of graphs and graph invariants, chromatic number, connectivity, planar graphs and conditions for planarity, statements of Four Color Theorem and Kuratowski's theorem, Hamiltonian cycles.<br /> <br /> Additional topics from 1) proof of Kuratowski's theorem, 2) definition of the Euler characteristic and proof of Euler's formula for the genus, 3) proof of the Chvatal-Erdos theorem on the existence of Hamiltonian cycles.<br /> <div style="-moz-column-count:2; column-count:2;"><br /> <br /> </div><br /> <br /> === Textbooks ===<br /> Possible textbooks for this course include (but are not limited to):<br /> <br /> * ''Graph Theory'' by Rusell Merris (Chap. 1-5).<br /> <br /> A good additional resource is An Introduction to Graph Theory by Douglas B. West, but the book is probably too encyclopedic to use as a main text.<br /> <br /> === Additional topics ===<br /> <br /> <br /> Possible other topics include spectral graph theory (networking, expanders, Ramanujan graphs), characterization of Ramanujan graphs by the Riemann hypothesis of its zeta function.<br /> " A good source for this topic is the paper by M. Ram Murty, Ramanujan Graphs, J. Ramanujan Math. Soc. 18(2003) 1-20.<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|355]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_355:_Graph_Theory&diff=3647 Math 355: Graph Theory 2017-11-09T18:08:35Z

<p>Jsc3: /* Additional topics */</p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Graph Theory.<br /> <br /> === (Credit Hours:Lecture Hours:Lab Hours) ===<br /> (3:3:0)<br /> <br /> === Offered ===<br /> W<br /> <br /> === Prerequisite ===<br /> [[Math 313]].<br /> <br /> === Description ===<br /> Graphs and subgraphs, coloring problems, applications.<br /> <br /> == Desired Learning Outcomes ==<br /> <br /> === Prerequisites ===<br /> <br /> Math 313.<br /> <br /> === Minimal learning outcomes ===<br /> Definition of a graph, examples (trees, paths, cycles, Petersen graph, etc.), First Theorem of Graph Theory, isomorphisms of graphs and graph invariants, chromatic number, connectivity, planar graphs and conditions for planarity, statements of Four Color Theorem and Kuratowski's theorem, Hamiltonian cycles.<br /> <br /> Additional topics from 1) proof of Kuratowski's theorem, 2) definition of the Euler characteristic and proof of Euler's formula for the genus, 3) proof of the Chvatal-Erdos theorem on the existence of Hamiltonian cycles.<br /> <div style="-moz-column-count:2; column-count:2;"><br /> <br /> </div><br /> <br /> === Textbooks ===<br /> Possible textbooks for this course include (but are not limited to):<br /> <br /> * ''Graph Theory'' by Rusell Merris (Chap. 1-5).<br /> <br /> A good additional resource is An Introduction to Graph Theory by Douglas B. West, but the book is probably too encyclopedic to use as a main text.<br /> <br /> === Additional topics ===<br /> <br /> <br /> Possible other topics include spectral graph theory (networking, expanders, Ramanujan graphs), characterization of Ramanujan graphs by the Riemann hypothesis of its zeta function.<br /> A good source for this topic is the paper by M. Ram Murty, Ramanujan Graphs, J. Ramanujan Math. Soc. 18(2003) 1-20.<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|355]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_355:_Graph_Theory&diff=3646 Math 355: Graph Theory 2017-11-09T18:07:20Z

<p>Jsc3: /* Additional topics */</p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Graph Theory.<br /> <br /> === (Credit Hours:Lecture Hours:Lab Hours) ===<br /> (3:3:0)<br /> <br /> === Offered ===<br /> W<br /> <br /> === Prerequisite ===<br /> [[Math 313]].<br /> <br /> === Description ===<br /> Graphs and subgraphs, coloring problems, applications.<br /> <br /> == Desired Learning Outcomes ==<br /> <br /> === Prerequisites ===<br /> <br /> Math 313.<br /> <br /> === Minimal learning outcomes ===<br /> Definition of a graph, examples (trees, paths, cycles, Petersen graph, etc.), First Theorem of Graph Theory, isomorphisms of graphs and graph invariants, chromatic number, connectivity, planar graphs and conditions for planarity, statements of Four Color Theorem and Kuratowski's theorem, Hamiltonian cycles.<br /> <br /> Additional topics from 1) proof of Kuratowski's theorem, 2) definition of the Euler characteristic and proof of Euler's formula for the genus, 3) proof of the Chvatal-Erdos theorem on the existence of Hamiltonian cycles.<br /> <div style="-moz-column-count:2; column-count:2;"><br /> <br /> </div><br /> <br /> === Textbooks ===<br /> Possible textbooks for this course include (but are not limited to):<br /> <br /> * ''Graph Theory'' by Rusell Merris (Chap. 1-5).<br /> <br /> A good additional resource is An Introduction to Graph Theory by Douglas B. West, but the book is probably too encyclopedic to use as a main text.<br /> <br /> === Additional topics ===<br /> <br /> <br /> Possible other topics include spectral graph theory (networking, expanders, Ramanujan graphs), characterization of Ramanujan graphs by the Riemann hypothesis of its zeta function.<br /> . A good source for this topic is the paper by M. Ram Murty, Ramanujan Graphs, J. Ramanujan Math. Soc. 18(2003) 1-20.<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|355]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_355:_Graph_Theory&diff=3645 Math 355: Graph Theory 2017-11-09T18:06:17Z

<p>Jsc3: /* Minimal learning outcomes */</p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Graph Theory.<br /> <br /> === (Credit Hours:Lecture Hours:Lab Hours) ===<br /> (3:3:0)<br /> <br /> === Offered ===<br /> W<br /> <br /> === Prerequisite ===<br /> [[Math 313]].<br /> <br /> === Description ===<br /> Graphs and subgraphs, coloring problems, applications.<br /> <br /> == Desired Learning Outcomes ==<br /> <br /> === Prerequisites ===<br /> <br /> Math 313.<br /> <br /> === Minimal learning outcomes ===<br /> Definition of a graph, examples (trees, paths, cycles, Petersen graph, etc.), First Theorem of Graph Theory, isomorphisms of graphs and graph invariants, chromatic number, connectivity, planar graphs and conditions for planarity, statements of Four Color Theorem and Kuratowski's theorem, Hamiltonian cycles.<br /> <br /> Additional topics from 1) proof of Kuratowski's theorem, 2) definition of the Euler characteristic and proof of Euler's formula for the genus, 3) proof of the Chvatal-Erdos theorem on the existence of Hamiltonian cycles.<br /> <div style="-moz-column-count:2; column-count:2;"><br /> <br /> </div><br /> <br /> === Textbooks ===<br /> Possible textbooks for this course include (but are not limited to):<br /> <br /> * ''Graph Theory'' by Rusell Merris (Chap. 1-5).<br /> <br /> A good additional resource is An Introduction to Graph Theory by Douglas B. West, but the book is probably too encyclopedic to use as a main text.<br /> <br /> === Additional topics ===<br /> <br /> <br /> Possible other topics include spectral graph theory (networking, expanders, Ramanujan graphs). A good source for this topic is the paper by M. Ram Murty, Ramanujan Graphs, J. Ramanujan Math. Soc. 18(2003) 1-20.<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|355]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_355:_Graph_Theory&diff=3644 Math 355: Graph Theory 2017-11-09T18:02:09Z

<p>Jsc3: /* Textbooks */</p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Graph Theory.<br /> <br /> === (Credit Hours:Lecture Hours:Lab Hours) ===<br /> (3:3:0)<br /> <br /> === Offered ===<br /> W<br /> <br /> === Prerequisite ===<br /> [[Math 313]].<br /> <br /> === Description ===<br /> Graphs and subgraphs, coloring problems, applications.<br /> <br /> == Desired Learning Outcomes ==<br /> <br /> === Prerequisites ===<br /> <br /> Math 313.<br /> <br /> === Minimal learning outcomes ===<br /> Definition of a graph, examples (trees, paths, cycles, Petersen graph, etc.), First Theorem of Graph Theory, isomorphisms of graphs and graph invariants, chromatic number, connectivity, planar graphs and conditions for planarity, statements of Four Color Theorem and Kuratowski's theorem, Hamiltonian cycles.<br /> <br /> Additional topics from 1) proof of Kuratowski's theorem, 2) definition of the Euler characteristic and proof of Euler's formula for the genus, 3) proof of the Chvatal-Erdos theorem on the existence of Hamiltonian cycles, 4) characterization of Ramanujan graphs by the Riemann hypothesis of its zeta function.<br /> <br /> <div style="-moz-column-count:2; column-count:2;"><br /> <br /> </div><br /> <br /> === Textbooks ===<br /> Possible textbooks for this course include (but are not limited to):<br /> <br /> * ''Graph Theory'' by Rusell Merris (Chap. 1-5).<br /> <br /> A good additional resource is An Introduction to Graph Theory by Douglas B. West, but the book is probably too encyclopedic to use as a main text.<br /> <br /> === Additional topics ===<br /> <br /> <br /> Possible other topics include spectral graph theory (networking, expanders, Ramanujan graphs). A good source for this topic is the paper by M. Ram Murty, Ramanujan Graphs, J. Ramanujan Math. Soc. 18(2003) 1-20.<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|355]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_355:_Graph_Theory&diff=3643 Math 355: Graph Theory 2017-11-09T18:01:03Z

<p>Jsc3: /* Minimal learning outcomes */</p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Graph Theory.<br /> <br /> === (Credit Hours:Lecture Hours:Lab Hours) ===<br /> (3:3:0)<br /> <br /> === Offered ===<br /> W<br /> <br /> === Prerequisite ===<br /> [[Math 313]].<br /> <br /> === Description ===<br /> Graphs and subgraphs, coloring problems, applications.<br /> <br /> == Desired Learning Outcomes ==<br /> <br /> === Prerequisites ===<br /> <br /> Math 313.<br /> <br /> === Minimal learning outcomes ===<br /> Definition of a graph, examples (trees, paths, cycles, Petersen graph, etc.), First Theorem of Graph Theory, isomorphisms of graphs and graph invariants, chromatic number, connectivity, planar graphs and conditions for planarity, statements of Four Color Theorem and Kuratowski's theorem, Hamiltonian cycles.<br /> <br /> Additional topics from 1) proof of Kuratowski's theorem, 2) definition of the Euler characteristic and proof of Euler's formula for the genus, 3) proof of the Chvatal-Erdos theorem on the existence of Hamiltonian cycles, 4) characterization of Ramanujan graphs by the Riemann hypothesis of its zeta function.<br /> <br /> <div style="-moz-column-count:2; column-count:2;"><br /> <br /> </div><br /> <br /> === Textbooks ===<br /> Possible textbooks for this course include (but are not limited to):<br /> <br /> * ''Graph Theory'' by Rusell Merris.<br /> <br /> A good additional resource is An Introduction to Graph Theory by Douglas B. West, but the book is probably too encyclopedic to use as a main text.<br /> <br /> === Additional topics ===<br /> <br /> <br /> Possible other topics include spectral graph theory (networking, expanders, Ramanujan graphs). A good source for this topic is the paper by M. Ram Murty, Ramanujan Graphs, J. Ramanujan Math. Soc. 18(2003) 1-20.<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|355]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_355:_Graph_Theory&diff=3642 Math 355: Graph Theory 2017-11-09T18:00:17Z

<p>Jsc3: /* Minimal learning outcomes */</p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Graph Theory.<br /> <br /> === (Credit Hours:Lecture Hours:Lab Hours) ===<br /> (3:3:0)<br /> <br /> === Offered ===<br /> W<br /> <br /> === Prerequisite ===<br /> [[Math 313]].<br /> <br /> === Description ===<br /> Graphs and subgraphs, coloring problems, applications.<br /> <br /> == Desired Learning Outcomes ==<br /> <br /> === Prerequisites ===<br /> <br /> Math 313.<br /> <br /> === Minimal learning outcomes ===<br /> Definition of a graph, examples (trees, paths, cycles, Petersen graph, etc.), First Theorem of Graph Theory, isomorphisms of graphs and graph invariants, chromatic number, connectivity, planar graphs and conditions for planarity, statements of Four Color Theorem and Kuratowski's theorem, Hamiltonian cycles.<br /> <br /> Additional topics from 1) proof of Kuratowski's theorem, 2) definition of the Euler characteristic and proof of the characteristic formula for the genus, 3) proof of the Chvatal-Erdos theorem on the existence of Hamiltonian cycles, 4) characterization of Ramanujan graphs by the Riemann hypothesis of its zeta function.<br /> <br /> <div style="-moz-column-count:2; column-count:2;"><br /> <br /> </div><br /> <br /> === Textbooks ===<br /> Possible textbooks for this course include (but are not limited to):<br /> <br /> * ''Graph Theory'' by Rusell Merris.<br /> <br /> A good additional resource is An Introduction to Graph Theory by Douglas B. West, but the book is probably too encyclopedic to use as a main text.<br /> <br /> === Additional topics ===<br /> <br /> <br /> Possible other topics include spectral graph theory (networking, expanders, Ramanujan graphs). A good source for this topic is the paper by M. Ram Murty, Ramanujan Graphs, J. Ramanujan Math. Soc. 18(2003) 1-20.<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|355]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_355:_Graph_Theory&diff=3641 Math 355: Graph Theory 2017-11-09T17:58:45Z

<p>Jsc3: /* Textbooks */</p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Graph Theory.<br /> <br /> === (Credit Hours:Lecture Hours:Lab Hours) ===<br /> (3:3:0)<br /> <br /> === Offered ===<br /> W<br /> <br /> === Prerequisite ===<br /> [[Math 313]].<br /> <br /> === Description ===<br /> Graphs and subgraphs, coloring problems, applications.<br /> <br /> == Desired Learning Outcomes ==<br /> <br /> === Prerequisites ===<br /> <br /> Math 313.<br /> <br /> === Minimal learning outcomes ===<br /> Definition of a graph, examples (trees, paths, cycles, Petersen graph, etc.), First Theorem of Graph Theory, isomorphisms of graphs and graph invariants, chromatic number, connectivity, planar graphs and conditions for planarity, statements of Four Color Theorem and Kuratowski's theorem, Hamiltonian cycles.<br /> <br /> Additional topics from 1) proof of Kuratowski's theorem, 2) definition of the Euler characteristic and proof of Euler's characteristic formula for the genus, 3) proof of the Chvatal-Erdos theorem on the existence of Hamiltonian cycles, 4) characterization of Ramanujan graphs by the Riemann hypothesis of its zeta function.<br /> <br /> <div style="-moz-column-count:2; column-count:2;"><br /> <br /> </div><br /> <br /> === Textbooks ===<br /> Possible textbooks for this course include (but are not limited to):<br /> <br /> * ''Graph Theory'' by Rusell Merris.<br /> <br /> A good additional resource is An Introduction to Graph Theory by Douglas B. West, but the book is probably too encyclopedic to use as a main text.<br /> <br /> === Additional topics ===<br /> <br /> <br /> Possible other topics include spectral graph theory (networking, expanders, Ramanujan graphs). A good source for this topic is the paper by M. Ram Murty, Ramanujan Graphs, J. Ramanujan Math. Soc. 18(2003) 1-20.<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|355]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_355:_Graph_Theory&diff=3640 Math 355: Graph Theory 2017-11-09T17:57:03Z

<p>Jsc3: /* Minimal learning outcomes */</p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Graph Theory.<br /> <br /> === (Credit Hours:Lecture Hours:Lab Hours) ===<br /> (3:3:0)<br /> <br /> === Offered ===<br /> W<br /> <br /> === Prerequisite ===<br /> [[Math 313]].<br /> <br /> === Description ===<br /> Graphs and subgraphs, coloring problems, applications.<br /> <br /> == Desired Learning Outcomes ==<br /> <br /> === Prerequisites ===<br /> <br /> Math 313.<br /> <br /> === Minimal learning outcomes ===<br /> Definition of a graph, examples (trees, paths, cycles, Petersen graph, etc.), First Theorem of Graph Theory, isomorphisms of graphs and graph invariants, chromatic number, connectivity, planar graphs and conditions for planarity, statements of Four Color Theorem and Kuratowski's theorem, Hamiltonian cycles.<br /> <br /> Additional topics from 1) proof of Kuratowski's theorem, 2) definition of the Euler characteristic and proof of Euler's characteristic formula for the genus, 3) proof of the Chvatal-Erdos theorem on the existence of Hamiltonian cycles, 4) characterization of Ramanujan graphs by the Riemann hypothesis of its zeta function.<br /> <br /> <div style="-moz-column-count:2; column-count:2;"><br /> <br /> </div><br /> <br /> === Textbooks ===<br /> Possible textbooks for this course include (but are not limited to):<br /> <br /> * ''Graph Theory'' by Rusell Merris.<br /> <br /> A good additional resource is An Introduction to Graph Theory by Douglas B. West, but the book is probably too encyclopedic to use as a main text.<br /> <br /> A good source for spectral graph theory is the paper by M. Ram Murty, Ramanujan Graphs, J. Ramanujan Math. Soc. 18(2003) 1-20.<br /> <br /> === Additional topics ===<br /> <br /> <br /> Possible other topics include spectral graph theory (networking, expanders, Ramanujan graphs). A good source for this topic is the paper by M. Ram Murty, Ramanujan Graphs, J. Ramanujan Math. Soc. 18(2003) 1-20.<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|355]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_355:_Graph_Theory&diff=3639 Math 355: Graph Theory 2017-11-09T17:55:51Z

<p>Jsc3: /* Description */</p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Graph Theory.<br /> <br /> === (Credit Hours:Lecture Hours:Lab Hours) ===<br /> (3:3:0)<br /> <br /> === Offered ===<br /> W<br /> <br /> === Prerequisite ===<br /> [[Math 313]].<br /> <br /> === Description ===<br /> Graphs and subgraphs, coloring problems, applications.<br /> <br /> == Desired Learning Outcomes ==<br /> <br /> === Prerequisites ===<br /> <br /> Math 313.<br /> <br /> === Minimal learning outcomes ===<br /> Definition of a graph, examples (trees, paths, cycles, Petersen graph, etc.), First Theorem of Graph Theory, isomorphisms of graphs and graph invariants, chromatic number, connectivity, planar graphs and conditions for planarity, statements of Four Color Theorem and Kuratowski's theorem, Hamiltonian cycles, spectral graph theory (networking, expanders, Ramanujan graphs).<br /> <br /> Additional topics from 1) proof of Kuratowski's theorem, 2) definition of the Euler characteristic and proof of Euler's characteristic formula for the genus, 3) proof of the Chvatal-Erdos theorem on the existence of Hamiltonian cycles, 4) characterization of Ramanujan graphs by the Riemann hypothesis of its zeta function.<br /> <br /> <div style="-moz-column-count:2; column-count:2;"><br /> <br /> </div><br /> <br /> === Textbooks ===<br /> Possible textbooks for this course include (but are not limited to):<br /> <br /> * ''Graph Theory'' by Rusell Merris.<br /> <br /> A good additional resource is An Introduction to Graph Theory by Douglas B. West, but the book is probably too encyclopedic to use as a main text.<br /> <br /> A good source for spectral graph theory is the paper by M. Ram Murty, Ramanujan Graphs, J. Ramanujan Math. Soc. 18(2003) 1-20.<br /> <br /> === Additional topics ===<br /> <br /> <br /> Possible other topics include spectral graph theory (networking, expanders, Ramanujan graphs). A good source for this topic is the paper by M. Ram Murty, Ramanujan Graphs, J. Ramanujan Math. Soc. 18(2003) 1-20.<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|355]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_355:_Graph_Theory&diff=3638 Math 355: Graph Theory 2017-11-09T17:53:01Z

<p>Jsc3: /* Additional topics */</p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Graph Theory.<br /> <br /> === (Credit Hours:Lecture Hours:Lab Hours) ===<br /> (3:3:0)<br /> <br /> === Offered ===<br /> W<br /> <br /> === Prerequisite ===<br /> [[Math 313]].<br /> <br /> === Description ===<br /> Maps, graphs and digraphs, coloring problems, applications.<br /> <br /> == Desired Learning Outcomes ==<br /> <br /> === Prerequisites ===<br /> <br /> Math 313.<br /> <br /> === Minimal learning outcomes ===<br /> Definition of a graph, examples (trees, paths, cycles, Petersen graph, etc.), First Theorem of Graph Theory, isomorphisms of graphs and graph invariants, chromatic number, connectivity, planar graphs and conditions for planarity, statements of Four Color Theorem and Kuratowski's theorem, Hamiltonian cycles, spectral graph theory (networking, expanders, Ramanujan graphs).<br /> <br /> Additional topics from 1) proof of Kuratowski's theorem, 2) definition of the Euler characteristic and proof of Euler's characteristic formula for the genus, 3) proof of the Chvatal-Erdos theorem on the existence of Hamiltonian cycles, 4) characterization of Ramanujan graphs by the Riemann hypothesis of its zeta function.<br /> <br /> <div style="-moz-column-count:2; column-count:2;"><br /> <br /> </div><br /> <br /> === Textbooks ===<br /> Possible textbooks for this course include (but are not limited to):<br /> <br /> * ''Graph Theory'' by Rusell Merris.<br /> <br /> A good additional resource is An Introduction to Graph Theory by Douglas B. West, but the book is probably too encyclopedic to use as a main text.<br /> <br /> A good source for spectral graph theory is the paper by M. Ram Murty, Ramanujan Graphs, J. Ramanujan Math. Soc. 18(2003) 1-20.<br /> <br /> === Additional topics ===<br /> <br /> <br /> Possible other topics include spectral graph theory (networking, expanders, Ramanujan graphs). A good source for this topic is the paper by M. Ram Murty, Ramanujan Graphs, J. Ramanujan Math. Soc. 18(2003) 1-20.<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|355]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_355:_Graph_Theory&diff=3637 Math 355: Graph Theory 2017-11-09T17:52:25Z

<p>Jsc3: /* Additional topics */</p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Graph Theory.<br /> <br /> === (Credit Hours:Lecture Hours:Lab Hours) ===<br /> (3:3:0)<br /> <br /> === Offered ===<br /> W<br /> <br /> === Prerequisite ===<br /> [[Math 313]].<br /> <br /> === Description ===<br /> Maps, graphs and digraphs, coloring problems, applications.<br /> <br /> == Desired Learning Outcomes ==<br /> <br /> === Prerequisites ===<br /> <br /> Math 313.<br /> <br /> === Minimal learning outcomes ===<br /> Definition of a graph, examples (trees, paths, cycles, Petersen graph, etc.), First Theorem of Graph Theory, isomorphisms of graphs and graph invariants, chromatic number, connectivity, planar graphs and conditions for planarity, statements of Four Color Theorem and Kuratowski's theorem, Hamiltonian cycles, spectral graph theory (networking, expanders, Ramanujan graphs).<br /> <br /> Additional topics from 1) proof of Kuratowski's theorem, 2) definition of the Euler characteristic and proof of Euler's characteristic formula for the genus, 3) proof of the Chvatal-Erdos theorem on the existence of Hamiltonian cycles, 4) characterization of Ramanujan graphs by the Riemann hypothesis of its zeta function.<br /> <br /> <div style="-moz-column-count:2; column-count:2;"><br /> <br /> </div><br /> <br /> === Textbooks ===<br /> Possible textbooks for this course include (but are not limited to):<br /> <br /> * ''Graph Theory'' by Rusell Merris.<br /> <br /> A good additional resource is An Introduction to Graph Theory by Douglas B. West, but the book is probably too encyclopedic to use as a main text.<br /> <br /> A good source for spectral graph theory is the paper by M. Ram Murty, Ramanujan Graphs, J. Ramanujan Math. Soc. 18(2003) 1-20.<br /> <br /> === Additional topics ===<br /> , spectral graph theory (networking, expanders, Ramanujan graphs).<br /> <br /> Possible other topics include spectral graph theory (networking, expanders, Ramanujan graphs). A good source for this topic is the paper by M. Ram Murty, Ramanujan Graphs, J. Ramanujan Math. Soc. 18(2003) 1-20.<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|355]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_355:_Graph_Theory&diff=3636 Math 355: Graph Theory 2017-11-09T17:51:14Z

<p>Jsc3: /* Additional topics */</p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Graph Theory.<br /> <br /> === (Credit Hours:Lecture Hours:Lab Hours) ===<br /> (3:3:0)<br /> <br /> === Offered ===<br /> W<br /> <br /> === Prerequisite ===<br /> [[Math 313]].<br /> <br /> === Description ===<br /> Maps, graphs and digraphs, coloring problems, applications.<br /> <br /> == Desired Learning Outcomes ==<br /> <br /> === Prerequisites ===<br /> <br /> Math 313.<br /> <br /> === Minimal learning outcomes ===<br /> Definition of a graph, examples (trees, paths, cycles, Petersen graph, etc.), First Theorem of Graph Theory, isomorphisms of graphs and graph invariants, chromatic number, connectivity, planar graphs and conditions for planarity, statements of Four Color Theorem and Kuratowski's theorem, Hamiltonian cycles, spectral graph theory (networking, expanders, Ramanujan graphs).<br /> <br /> Additional topics from 1) proof of Kuratowski's theorem, 2) definition of the Euler characteristic and proof of Euler's characteristic formula for the genus, 3) proof of the Chvatal-Erdos theorem on the existence of Hamiltonian cycles, 4) characterization of Ramanujan graphs by the Riemann hypothesis of its zeta function.<br /> <br /> <div style="-moz-column-count:2; column-count:2;"><br /> <br /> </div><br /> <br /> === Textbooks ===<br /> Possible textbooks for this course include (but are not limited to):<br /> <br /> * ''Graph Theory'' by Rusell Merris.<br /> <br /> A good additional resource is An Introduction to Graph Theory by Douglas B. West, but the book is probably too encyclopedic to use as a main text.<br /> <br /> A good source for spectral graph theory is the paper by M. Ram Murty, Ramanujan Graphs, J. Ramanujan Math. Soc. 18(2003) 1-20.<br /> <br /> === Additional topics ===<br /> , spectral graph theory (networking, expanders, Ramanujan graphs).<br /> <br /> Possible other topics include spectral graph theory (networking, expanders, Ramanujan graphs).<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|355]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_355:_Graph_Theory&diff=3635 Math 355: Graph Theory 2017-11-09T17:47:14Z

<p>Jsc3: /* Textbooks */</p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Graph Theory.<br /> <br /> === (Credit Hours:Lecture Hours:Lab Hours) ===<br /> (3:3:0)<br /> <br /> === Offered ===<br /> W<br /> <br /> === Prerequisite ===<br /> [[Math 313]].<br /> <br /> === Description ===<br /> Maps, graphs and digraphs, coloring problems, applications.<br /> <br /> == Desired Learning Outcomes ==<br /> <br /> === Prerequisites ===<br /> <br /> Math 313.<br /> <br /> === Minimal learning outcomes ===<br /> Definition of a graph, examples (trees, paths, cycles, Petersen graph, etc.), First Theorem of Graph Theory, isomorphisms of graphs and graph invariants, chromatic number, connectivity, planar graphs and conditions for planarity, statements of Four Color Theorem and Kuratowski's theorem, Hamiltonian cycles, spectral graph theory (networking, expanders, Ramanujan graphs).<br /> <br /> Additional topics from 1) proof of Kuratowski's theorem, 2) definition of the Euler characteristic and proof of Euler's characteristic formula for the genus, 3) proof of the Chvatal-Erdos theorem on the existence of Hamiltonian cycles, 4) characterization of Ramanujan graphs by the Riemann hypothesis of its zeta function.<br /> <br /> <div style="-moz-column-count:2; column-count:2;"><br /> <br /> </div><br /> <br /> === Textbooks ===<br /> Possible textbooks for this course include (but are not limited to):<br /> <br /> * ''Graph Theory'' by Rusell Merris.<br /> <br /> A good additional resource is An Introduction to Graph Theory by Douglas B. West, but the book is probably too encyclopedic to use as a main text.<br /> <br /> A good source for spectral graph theory is the paper by M. Ram Murty, Ramanujan Graphs, J. Ramanujan Math. Soc. 18(2003) 1-20.<br /> <br /> === Additional topics ===<br /> Possible other topics include spectral theory of graphs.<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|355]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_355:_Graph_Theory&diff=3634 Math 355: Graph Theory 2017-11-09T17:46:41Z

<p>Jsc3: /* Textbooks */</p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Graph Theory.<br /> <br /> === (Credit Hours:Lecture Hours:Lab Hours) ===<br /> (3:3:0)<br /> <br /> === Offered ===<br /> W<br /> <br /> === Prerequisite ===<br /> [[Math 313]].<br /> <br /> === Description ===<br /> Maps, graphs and digraphs, coloring problems, applications.<br /> <br /> == Desired Learning Outcomes ==<br /> <br /> === Prerequisites ===<br /> <br /> Math 313.<br /> <br /> === Minimal learning outcomes ===<br /> Definition of a graph, examples (trees, paths, cycles, Petersen graph, etc.), First Theorem of Graph Theory, isomorphisms of graphs and graph invariants, chromatic number, connectivity, planar graphs and conditions for planarity, statements of Four Color Theorem and Kuratowski's theorem, Hamiltonian cycles, spectral graph theory (networking, expanders, Ramanujan graphs).<br /> <br /> Additional topics from 1) proof of Kuratowski's theorem, 2) definition of the Euler characteristic and proof of Euler's characteristic formula for the genus, 3) proof of the Chvatal-Erdos theorem on the existence of Hamiltonian cycles, 4) characterization of Ramanujan graphs by the Riemann hypothesis of its zeta function.<br /> <br /> <div style="-moz-column-count:2; column-count:2;"><br /> <br /> </div><br /> <br /> === Textbooks ===<br /> Possible textbooks for this course include (but are not limited to):<br /> <br /> * ''Graph Theory'' by Rusell Merris.<br /> <br /> A good additional resource is An Introduction to Graph Theory by Douglas B. West, but the book is probably too encyclopedic to use as a main text.<br /> <br /> * ''A good source for spectral graph theory is the paper by M. Ram Murty, Ramanujan Graphs, J. Ramanujan Math. Soc. 18(2003) 1-20.<br /> <br /> === Additional topics ===<br /> Possible other topics include spectral theory of graphs.<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|355]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_355:_Graph_Theory&diff=3633 Math 355: Graph Theory 2017-11-09T17:28:18Z

<p>Jsc3: /* Minimal learning outcomes */</p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Graph Theory.<br /> <br /> === (Credit Hours:Lecture Hours:Lab Hours) ===<br /> (3:3:0)<br /> <br /> === Offered ===<br /> W<br /> <br /> === Prerequisite ===<br /> [[Math 313]].<br /> <br /> === Description ===<br /> Maps, graphs and digraphs, coloring problems, applications.<br /> <br /> == Desired Learning Outcomes ==<br /> <br /> === Prerequisites ===<br /> <br /> Math 313.<br /> <br /> === Minimal learning outcomes ===<br /> Definition of a graph, examples (trees, paths, cycles, Petersen graph, etc.), First Theorem of Graph Theory, isomorphisms of graphs and graph invariants, chromatic number, connectivity, planar graphs and conditions for planarity, statements of Four Color Theorem and Kuratowski's theorem, Hamiltonian cycles, spectral graph theory (networking, expanders, Ramanujan graphs).<br /> <br /> Additional topics from 1) proof of Kuratowski's theorem, 2) definition of the Euler characteristic and proof of Euler's characteristic formula for the genus, 3) proof of the Chvatal-Erdos theorem on the existence of Hamiltonian cycles, 4) characterization of Ramanujan graphs by the Riemann hypothesis of its zeta function.<br /> <br /> <div style="-moz-column-count:2; column-count:2;"><br /> <br /> </div><br /> <br /> === Textbooks ===<br /> Possible textbooks for this course include (but are not limited to):<br /> <br /> * ''Graph Theory'' by Rusell Merris<br /> <br /> * ''Applied Combinatorics'' by Alan Tucker (Chapters 1-4 on Graph Theory).<br /> <br /> A good additional resource is An Introduction to Graph Theory by Douglas B. West, but the book is probably too encyclopedic to use as a main text.<br /> <br /> === Additional topics ===<br /> Possible other topics include spectral theory of graphs.<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|355]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_686R:_Topics_in_Algebraic_Number_Theory.&diff=1616 Math 686R: Topics in Algebraic Number Theory. 2010-11-01T14:58:38Z

<p>Jsc3: </p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Topics in Algebraic Number Theory.<br /> <br /> === Credit Hours ===<br /> 3<br /> <br /> === Prerequisite ===<br /> [[Math 352]], [[Math 487|487]], [[Math 671|671]], [[Math 672|672]].<br /> <br /> === Description ===<br /> Current topics of research interest.<br /> <br /> == Desired Learning Outcomes ==<br /> To gain familiarity with working in general settings, e.g. elliptic curves over number fields, and not just over rationals.<br /> === Prerequisites ===<br /> The pre-requisites in the Catalog are adequate. Depending on the instructor and with her/his permission, a sound understanding of basic concepts of complex analysis, abstract algebra and number theory might be adequate, perhaps at the level of Math 352, 371, 372 and 487.<br /> <br /> === Minimal learning outcomes ===<br /> <br /> <div style="-moz-column-count:2; column-count:2;"><br /> #Generalization of the Unique Factorization Theorem from rationals to number fields; Basic definitions: <br /> #* Number fields<br /> #* Algebraic integers in a number field<br /> #* Integral bases<br /> #* Discriminant<br /> #* Norms of ideals<br /> #* Finiteness of ideals of bounded norm<br /> #* Class number<br /> #* Finiteness of class number<br /> #* Dedekind's Unique Factorization Theorem for ideals of a number field<br /> #Geometry of numbers: <br /> #* Minkowski's lemma on lattice points<br /> #* Logarithmic spaces<br /> #* Dirichlet's Unit Theorem for the units of the ring of integers of a number field<br /> #* Theorems of Minkowski and of Hermite on discriminants of number fields<br /> #Ramification Theory: <br /> #* Relative extensions<br /> #* Relative discriminant and Dedekind's criterion for ramification in terms of discriminant<br /> #* Higher ramification groups<br /> #* Hilbert theory of ramification<br /> #Splitting of Primes: <br /> #* Frobenius map<br /> #* Artin symbol<br /> #* Artin map<br /> #* Splitting of primes in Abelian extensions in terms of Artin map<br /> #* Rudimentary class field theory<br /> #* Examples - quadratic and cyclotomic extensions<br /> #Arithmetic of cyclotomic fields, and the Kronecker-Weber Theorem<br /> #Dedekind zeta function<br /> #* The class number formula - the formula which relates the residue of the Dedekind zeta function of a number field at s=1 to its class number, regulator and discriminant<br /> </div><br /> <br /> <br /> In addition, time permitting, the instructor may want to add to the list topics of special interest to him/her.<br /> <br /> === Textbooks ===<br /> <br /> Possible textbooks for this course include, (but are not limited to): <br /> #* D. Marcus, Number Fields (Universitext)<br /> <br /> <br /> <br /> === Additional topics ===<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|686]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_686R:_Topics_in_Algebraic_Number_Theory.&diff=1601 Math 686R: Topics in Algebraic Number Theory. 2010-10-29T21:39:32Z

<p>Jsc3: </p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Topics in Algebraic Number Theory.<br /> <br /> === Credit Hours ===<br /> 3<br /> <br /> === Prerequisite ===<br /> [[Math 352]], [[Math 487|487]], [[Math 671|671]], [[Math 672|672]].<br /> <br /> === Description ===<br /> Current topics of research interest.<br /> <br /> == Desired Learning Outcomes ==<br /> To gain familiarity with working in general settings, e.g. Abelian varieties over number fields, not just of elliptic curves over rationals.<br /> === Prerequisites ===<br /> The pre-requisites in the Catalog are adequate. Depending on the instructor and with her/his permission, a sound understanding of basic concepts of complex analysis, abstract algebra and number theory might be adequate, perhaps at the level of Math 352, 371, 372 and 487.<br /> <br /> === Minimal learning outcomes ===<br /> <br /> <div style="-moz-column-count:2; column-count:2;"><br /> #Generalization of the Unique Factorization Theorem from rationals to number fields; Basic definitions: <br /> #* Number fields<br /> #* Algebraic integers in a number field<br /> #* Integral bases<br /> #* Discriminant<br /> #* Norms of ideals<br /> #* Finiteness of ideals of bounded norm<br /> #* Class number<br /> #* Finiteness of class number<br /> #* Dedekind's Unique Factorization Theorem for ideals of a number field<br /> #Geometry of numbers: <br /> #* Minkowski's lemma on lattice points<br /> #* Logarithmic spaces<br /> #* Dirichlet's Unit Theorem for the units of the ring of integers of a number field<br /> #* Theorems of Minkowski and of Hermite on discriminants of number fields<br /> #Ramification Theory: <br /> #* Relative extensions<br /> #* Relative discriminant and Dedekind's criterion for ramification in terms of discriminant<br /> #* Higher ramification groups<br /> #* Hilbert theory of ramification<br /> #Splitting of Primes: <br /> #* Frobenius map<br /> #* Artin symbol<br /> #* Artin map<br /> #* Splitting of primes in Abelian extensions in terms of Artin map<br /> #* Rudimentary class field theory<br /> #* Examples - quadratic and cyclotomic extensions<br /> #Arithmetic of cyclotomic fields, and the Kronecker-Weber Theorem<br /> #Dedekind zeta function<br /> #* The class number formula - the formula which relates the residue of the Dedekind zeta function of a number field at s=1 to its class number, regulator and discriminant<br /> </div><br /> <br /> <br /> In addition, time permitting, the instructor may want to add to the list topics of special interest to him/her.<br /> <br /> === Textbooks ===<br /> <br /> Possible textbooks for this course include, (but are not limited to): <br /> #* D. Marcus, Number Fields (Universitext)<br /> <br /> <br /> <br /> === Additional topics ===<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|686]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_686R:_Topics_in_Algebraic_Number_Theory.&diff=1600 Math 686R: Topics in Algebraic Number Theory. 2010-10-29T21:38:39Z

<p>Jsc3: </p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Topics in Algebraic Number Theory.<br /> <br /> === Credit Hours ===<br /> 3<br /> <br /> === Prerequisite ===<br /> [[Math 352]], [[Math 487|487]], [[Math 671|671]], [[Math 672|672]].<br /> <br /> === Description ===<br /> Current topics of research interest.<br /> <br /> == Desired Learning Outcomes ==<br /> To gain familiarity with working in general settings, e.g. Abelian varieties over number fields, not just of elliptic curves over rationals.<br /> === Prerequisites ===<br /> The pre-requisites in the Catalog are adequate. Depending on the instructor and with her/his permission, a sound understanding of basic concepts of complex analysis, abstract algebra and number theory might be adequate, perhaps at the level of Math 352, 371, 372 and 487.<br /> <br /> === Minimal learning outcomes ===<br /> <br /> <div style="-moz-column-count:2; column-count:2;"><br /> #Generalization of the Unique Factorization Theorem from rationals to number fields; Basic definitions: <br /> #* Number fields<br /> #* Algebraic integers in a number field<br /> #* Integral bases<br /> #* Discriminant<br /> #* Norms of ideals<br /> #* Finiteness of ideals of bounded norm<br /> #* Class number<br /> #* Finiteness of class number<br /> #* Dedekind's Unique Factorization Theorem for ideals of a number field<br /> #Geometry of numbers: <br /> #* Minkowski's lemma on lattice points<br /> #* Logarithmic spaces<br /> #* Dirichlet's Unit Theorem for the units of the ring of integers of a number field<br /> #* Theorems of Minkowski and of Hermite on discriminants of number fields<br /> #Ramification Theory: <br /> #* Relative extensions<br /> #* Relative discriminant and Dedekind's criterion for ramification in terms of discriminant<br /> #* Higher ramification groups<br /> #* Hilbert theory of ramification<br /> #Splitting of Primes: <br /> #* Frobenius map<br /> #* Artin symbol<br /> #* Artin map<br /> #* Splitting of primes in Abelian extensions in terms of Artin map<br /> #* Rudimentary class field theory<br /> #* Examples - quadratic and cyclotomic extensions<br /> #Arithmetic of cyclotomic fields, and the Kronecker-Weber Theorem<br /> #Dedekind zeta function<br /> #* The class number formula - the formula which relates the residue of the Dedekind zeta function of a number field at s=1 to its class number, regulator and discriminant<br /> </div><br /> <br /> <br /> In addition, time permitting, the instructor may want to add to the list topics of special interest to him/her.<br /> <br /> === Textbooks ===<br /> <br /> Possible textbooks for this course include, (but are not limited to): <br /> #* D. Marcus, Number Fields (Universitext)<br /> <br /> *<br /> <br /> === Additional topics ===<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|686]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_686R:_Topics_in_Algebraic_Number_Theory.&diff=1599 Math 686R: Topics in Algebraic Number Theory. 2010-10-29T21:37:31Z

<p>Jsc3: </p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Topics in Algebraic Number Theory.<br /> <br /> === Credit Hours ===<br /> 3<br /> <br /> === Prerequisite ===<br /> [[Math 352]], [[Math 487|487]], [[Math 671|671]], [[Math 672|672]].<br /> <br /> === Description ===<br /> Current topics of research interest.<br /> <br /> == Desired Learning Outcomes ==<br /> To gain familiarity with working in general settings, e.g. Abelian varieties over number fields, not just of elliptic curves over rationals.<br /> === Prerequisites ===<br /> The pre-requisites in the Catalog are adequate. Depending on the instructor and with her/his permission, a sound understanding of basic concepts of complex analysis, abstract algebra and number theory might be adequate, perhaps at the level of Math 352, 371, 372 and 487.<br /> <br /> === Minimal learning outcomes ===<br /> <br /> <div style="-moz-column-count:2; column-count:2;"><br /> #Generalization of the Unique Factorization Theorem from rationals to number fields; Basic definitions: <br /> #* Number fields<br /> #* Algebraic integers in a number field<br /> #* Integral bases<br /> #* Discriminant<br /> #* Norms of ideals<br /> #* Finiteness of ideals of bounded norm<br /> #* Class number<br /> #* Finiteness of class number<br /> #* Dedekind's Unique Factorization Theorem for ideals of a number field<br /> #Geometry of numbers: <br /> #* Minkowski's lemma on lattice points<br /> #* Logarithmic spaces<br /> #* Dirichlet's Unit Theorem for the units of the ring of integers of a number field<br /> #* Theorems of Minkowski and of Hermite on discriminants of number fields<br /> #Ramification Theory: <br /> #* Relative extensions<br /> #* Relative discriminant and Dedekind's criterion for ramification in terms of discriminant<br /> #* Higher ramification groups<br /> #* Hilbert theory of ramification<br /> #Splitting of Primes: <br /> #* Frobenius map<br /> #* Artin symbol<br /> #* Artin map<br /> #* Splitting of primes in Abelian extensions in terms of Artin map<br /> #* Rudimentary class field theory<br /> #* Examples - quadratic and cyclotomic extensions<br /> #Arithmetic of cyclotomic fields, and the Kronecker-Weber Theorem<br /> #Dedekind zeta function<br /> #* The class number formula - the formula which relates the residue of the Dedekind zeta function of a number field at s=1 to its class number, regulator and discriminant<br /> </div><br /> <br /> <br /> In addition, time permitting, the instructor may want to add to the list topics of special interest to him/her.<br /> <br /> === Textbooks ===<br /> <br /> Possible textbooks for this course include, (but are not limited to): <br /> <br /> #* D. Marcus, Number Fields (Universitext)<br /> <br /> *<br /> <br /> === Additional topics ===<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|686]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_686R:_Topics_in_Algebraic_Number_Theory.&diff=1598 Math 686R: Topics in Algebraic Number Theory. 2010-10-29T21:36:24Z

<p>Jsc3: </p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Topics in Algebraic Number Theory.<br /> <br /> === Credit Hours ===<br /> 3<br /> <br /> === Prerequisite ===<br /> [[Math 352]], [[Math 487|487]], [[Math 671|671]], [[Math 672|672]].<br /> <br /> === Description ===<br /> Current topics of research interest.<br /> <br /> == Desired Learning Outcomes ==<br /> To gain familiarity with working in general settings, e.g. Abelian varieties over number fields, not just of elliptic curves over rationals.<br /> === Prerequisites ===<br /> The pre-requisites in the Catalog are adequate. Depending on the instructor and with her/his permission, a sound understanding of basic concepts of complex analysis, abstract algebra and number theory might be adequate, perhaps at the level of Math 352, 371, 372 and 487.<br /> <br /> === Minimal learning outcomes ===<br /> <br /> <div style="-moz-column-count:2; column-count:2;"><br /> #Generalization of the Unique Factorization Theorem from rationals to number fields; Basic definitions: <br /> #* Number fields<br /> #* Algebraic integers in a number field<br /> #* Integral bases<br /> #* Discriminant<br /> #* Norms of ideals<br /> #* Finiteness of ideals of bounded norm<br /> #* Class number<br /> #* Finiteness of class number<br /> #* Dedekind's Unique Factorization Theorem for ideals of a number field<br /> #Geometry of numbers: <br /> #* Minkowski's lemma on lattice points<br /> #* Logarithmic spaces<br /> #* Dirichlet's Unit Theorem for the units of the ring of integers of a number field<br /> #* Theorems of Minkowski and of Hermite on discriminants of number fields<br /> #Ramification Theory: <br /> #* Relative extensions<br /> #* Relative discriminant and Dedekind's criterion for ramification in terms of discriminant<br /> #* Higher ramification groups<br /> #* Hilbert theory of ramification<br /> #Splitting of Primes: <br /> #* Frobenius map<br /> #* Artin symbol<br /> #* Artin map<br /> #* Splitting of primes in Abelian extensions in terms of Artin map<br /> #* Rudimentary class field theory<br /> #* Examples - quadratic and cyclotomic extensions<br /> #Arithmetic of cyclotomic fields, and the Kronecker-Weber Theorem<br /> #Dedekind zeta function<br /> #* The class number formula - the formula which relates the residue of the Dedekind zeta function of a number field at s=1 to its class number, regulator and discriminant<br /> </div><br /> <br /> <br /> In addition, time permitting, the instructor may want to add to the list topics of special interest to him/her.<br /> <br /> === Textbooks ===<br /> <br /> Possible textbooks for this course include, (but are not limited to): <br /> <br /> D. Marcus, Number Fields (Universitext)<br /> <br /> *<br /> <br /> === Additional topics ===<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|686]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_686R:_Topics_in_Algebraic_Number_Theory.&diff=1597 Math 686R: Topics in Algebraic Number Theory. 2010-10-29T21:30:03Z

<p>Jsc3: </p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Topics in Algebraic Number Theory.<br /> <br /> === Credit Hours ===<br /> 3<br /> <br /> === Prerequisite ===<br /> [[Math 352]], [[Math 487|487]], [[Math 671|671]], [[Math 672|672]].<br /> <br /> === Description ===<br /> Current topics of research interest.<br /> <br /> == Desired Learning Outcomes ==<br /> To gain familiarity with working in general settings, e.g. Abelian varieties over number fields, not just of elliptic curves over rationals.<br /> === Prerequisites ===<br /> The pre-requisites in the Catalog are adequate. Depending on the instructor and with her/his permission, a sound understanding of basic concepts of complex analysis, abstract algebra and number theory might be adequate, perhaps at the level of Math 352, 371, 372 and 487.<br /> <br /> === Minimal learning outcomes ===<br /> <br /> <div style="-moz-column-count:2; column-count:2;"><br /> #Generalization of the Unique Factorization Theorem from rationals to number fields; Basic definitions: <br /> #* Number fields<br /> #* Algebraic integers in a number field<br /> #* Integral bases<br /> #* Discriminant<br /> #* Norms of ideals<br /> #* Finiteness of ideals of bounded norm<br /> #* Class number<br /> #* Finiteness of class number<br /> #* Dedekind's Unique Factorization Theorem for ideals of a number field<br /> #Geometry of numbers: <br /> #* Minkowski's lemma on lattice points<br /> #* Logarithmic spaces<br /> #* Dirichlet's Unit Theorem for the units of the ring of integers of a number field<br /> #* Theorems of Minkowski and of Hermite on discriminants of number fields<br /> #Ramification Theory: <br /> #* Relative extensions<br /> #* Relative discriminant and Dedekind's criterion for ramification in terms of discriminant<br /> #* Higher ramification groups<br /> #* Hilbert theory of ramification<br /> #Splitting of Primes: <br /> #* Frobenius map<br /> #* Artin symbol<br /> #* Artin map<br /> #* Splitting of primes in Abelian extensions in terms of Artin map<br /> #* Rudimentary class field theory<br /> #* Examples - quadratic and cyclotomic extensions<br /> #Arithmetic of cyclotomic fields, and the Kronecker-Weber Theorem<br /> #Dedekind zeta function<br /> #* The class number formula - the formula which relates the residue of the Dedekind zeta function of a number field at s=1 to its class number, regulator and discriminant<br /> </div><br /> <br /> <br /> In addition, time permitting, the instructor may want to add to the list topics of special interest to him/her.<br /> <br /> === Textbooks ===<br /> <br /> Possible textbooks for this course include, (but are not limited to): Marcus, Number Fields, Springer <br /> <br /> *<br /> <br /> === Additional topics ===<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|686]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_686R:_Topics_in_Algebraic_Number_Theory.&diff=1596 Math 686R: Topics in Algebraic Number Theory. 2010-10-29T21:24:49Z

<p>Jsc3: </p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Topics in Algebraic Number Theory.<br /> <br /> === Credit Hours ===<br /> 3<br /> <br /> === Prerequisite ===<br /> [[Math 352]], [[Math 487|487]], [[Math 671|671]], [[Math 672|672]].<br /> <br /> === Description ===<br /> Current topics of research interest.<br /> <br /> == Desired Learning Outcomes ==<br /> To gain familiarity with working in general settings, e.g. Abelian varieties over number fields, not just of elliptic curves over rationals.<br /> === Prerequisites ===<br /> The pre-requisites in the Catalog are adequate. Depending on the instructor and with her/his permission, a sound understanding of basic concepts of complex analysis, abstract algebra and number theory might be adequate, perhaps at the level of Math 352, 371, 372 and 487.<br /> <br /> === Minimal learning outcomes ===<br /> <br /> <div style="-moz-column-count:2; column-count:2;"><br /> #Generalization of the Unique Factorization Theorem from rationals to number fields; Basic definitions: <br /> #* Number fields<br /> #* Algebraic integers in a number field<br /> #* Integral bases<br /> #* Discriminant<br /> #* Norms of ideals<br /> #* Finiteness of ideals of bounded norm<br /> #* Class number<br /> #* Finiteness of class number<br /> #* Dedekind's Unique Factorization Theorem for ideals of a number field<br /> #Geometry of numbers: <br /> #* Minkowski's lemma on lattice points<br /> #* Logarithmic spaces<br /> #* Dirichlet's Unit Theorem for the units of the ring of integers of a number field<br /> #* Theorems of Minkowski and of Hermite on discriminants of number fields<br /> #Ramification Theory: <br /> #* Relative extensions<br /> #* Relative discriminant and Dedekind's criterion for ramification in terms of discriminant<br /> #* Higher ramification groups<br /> #* Hilbert theory of ramification<br /> #Splitting of Primes: <br /> #* Frobenius map<br /> #* Artin symbol<br /> #* Artin map<br /> #* Splitting of primes in Abelian extensions in terms of Artin map<br /> #* Rudimentary class field theory<br /> #* Examples - quadratic and cyclotomic extensions<br /> #Arithmetic of cyclotomic fields, and the Kronecker-Weber Theorem<br /> #Dedekind zeta function<br /> #* The class number formula - the formula which relates the residue of the Dedekind zeta function of a number field at s=1 to its class number, regulator and discriminant<br /> </div><br /> <br /> <br /> In addition, time permitting, the instructor may want to add to the list topics of special interest to him/her.<br /> <br /> === Textbooks ===<br /> <br /> Possible textbooks for this course include (but are not limited to):<br /> <br /> *<br /> <br /> === Additional topics ===<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|686]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_686R:_Topics_in_Algebraic_Number_Theory.&diff=1585 Math 686R: Topics in Algebraic Number Theory. 2010-10-29T15:22:02Z

<p>Jsc3: </p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Topics in Algebraic Number Theory.<br /> <br /> === Credit Hours ===<br /> 3<br /> <br /> === Prerequisite ===<br /> [[Math 352, 487, 671, 672]].<br /> <br /> === Description ===<br /> Current topics of research interest.<br /> <br /> == Desired Learning Outcomes ==<br /> To gain familiarity with working in general settings, e.g. Abelian varieties over number fields, not just of elliptic curves over rationals.<br /> === Prerequisites ===<br /> A sound understanding of basic concepts of complex analysis, abstract algebra and number theory, at the level of Math 352, 371, 372 and 487 may suffice.<br /> === Minimal learning outcomes ===<br /> <br /> 1. Generalization of the Unique Factorization Theorem from rationals to number fields: Basic Definitions - number fields, algebraic integers in a number field, integral bases, discriminant, norms of ideals, finiteness of ideals of bounded norm, class number, finiteness of class number, Dedekind's Unique Factorization Theorem for ideals of a number field.<br /> 2. Geometry of numbers: Minkowski's lemma on lattice points, Logarithmic spaces, Dirichlet's Unit Theorem for the units of the ring of integers of a number field, theorems of Minkowski and of Hermite on discriminants of number fields.<br /> 3. Ramification Theory: Relative extensions, relative discriminant and Dedekind's criterion for ramification in terms of discriminant, higher ramification groups and Hilbert theory of ramification.<br /> 4 Splitting of Primes: Frobenius map, Artin symbol, Artin map and splitting of primes in Abelian extensions in terms of Artin map, rudimentary class field theory, Examples - quadratic and cyclotomic extensions.<br /> 5. Arithmetic of cyclotomic fields, the Kronecker-Weber Theorem.<br /> 6. Dedekind zeta function, the class number formula - the formula which relates the residue of the Dedekind zeta function of a number field at s=1 to its class number, regulator and discriminant.<br /> <br /> In addition, time permitting, the instructor may want to add to the list topics of special to him/her.<br /> <br /> <br /> <div style="-moz-column-count:2; column-count:2;"><br /> <br /> </div><br /> === Textbooks ===<br /> <br /> Possible textbooks for this course include (but are not limited to):<br /> <br /> *<br /> <br /> === Additional topics ===<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|686]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_686R:_Topics_in_Algebraic_Number_Theory.&diff=1584 Math 686R: Topics in Algebraic Number Theory. 2010-10-29T15:19:57Z

<p>Jsc3: </p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Topics in Algebraic Number Theory.<br /> <br /> === Credit Hours ===<br /> 3<br /> <br /> === Prerequisite ===<br /> [[Math 352, 487, 671, 672]].<br /> <br /> === Description ===<br /> Current topics of research interest.<br /> <br /> == Desired Learning Outcomes ==<br /> To gain familiarity with working in general settings, e.g. Abelian varieties over number fields, not just of elliptic curves over rationals.<br /> === Prerequisites ===<br /> A sound understanding of basic concepts of complex analysis, abstract algebra and number theory, at the level of Math 352, 371, 372 and 487 may suffice.<br /> === Minimal learning outcomes ===<br /> <br /> 1. Generalization of the Unique Factorization Theorem from rationals to number fields: Basic Definitions - number fields, algebraic integers in a number field, integral bases, discriminant, norms of ideals, finiteness of ideals of bounded norm, class number, finiteness of class number, Dedekind's Unique Factorization Theorem for ideals of a number field.<br /> 2. Geometry of numbers: Minkowski's lemma on lattice points, Logarithmic spaces, Dirichlet's Unit Theorem for the units of the ring of integers of a number field, theorems of Minkowski and of Hermite on discriminants of number fields.<br /> 3. Ramification Theory: Relative extensions, relative discriminant and Dedekind's criterion for ramification in terms of discriminant, higher ramification groups and Hilbert theory of ramification.<br /> 4 Splitting of Primes: Frobenius map, Artin symbol, Artin map and splitting of primes in Abelian extensions in terms of Artin map, rudimentary class field theory, Examples - quadratic and cyclotomic extensions.<br /> 5. Arithmetic of cyclotomic fields, the Kronecker-Weber Theorem.<br /> 6. Dedekind zeta function, the class number formula, which gives the residue of the Dedekind zeta function of a number field at s=1 in terms of its class number, regulator and discriminant.<br /> <br /> In addition, time permitting, the instructor may want to add to the list topics of special to him/her.<br /> <br /> <br /> <div style="-moz-column-count:2; column-count:2;"><br /> <br /> </div><br /> === Textbooks ===<br /> <br /> Possible textbooks for this course include (but are not limited to):<br /> <br /> *<br /> <br /> === Additional topics ===<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|686]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_686R:_Topics_in_Algebraic_Number_Theory.&diff=1583 Math 686R: Topics in Algebraic Number Theory. 2010-10-29T01:11:48Z

<p>Jsc3: </p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Topics in Algebraic Number Theory.<br /> <br /> === Credit Hours ===<br /> 3<br /> <br /> === Prerequisite ===<br /> [[Math 352, 487, 671, 672]].<br /> <br /> === Description ===<br /> Current topics of research interest.<br /> <br /> == Desired Learning Outcomes ==<br /> To gain familiarity with working in general settings, e.g. Abelian varieties over number fields, not just of elliptic curves over rationals.<br /> === Prerequisites ===<br /> A sound understanding of basic concepts of complex analysis, abstract algebra and number theory, at the level of Math 352, 371, 372 and 487 may suffice.<br /> === Minimal learning outcomes ===<br /> <br /> 1. Generalization of the Unique Factorization Theorem from rationals to number fields: Basic Definitions - number fields, algebraic integers in a number field, integral bases, discriminant, norms of ideals, finiteness of ideals of bounded norm, class number, finiteness of class number, Dedekind's Unique Factorization Theorem for ideals of a number field.<br /> 2. Geometry of numbers: Minkowski's lemma on lattice points, Logarithmic spaces, Dirichlet's Unit Theorem for the units of the ring of integers of a number field, theorems of Minkowski and of Hermite on discriminants of number fields.<br /> 3. Ramification Theory: Relative extensions, relative discriminant and Dedekind's criterion for ramification in terms of discriminant, higher ramification groups and Hilbert theory of ramification.<br /> 4 Splitting of Primes: Frobenius map, Artin symbol, Artin map and splitting of primes in Abelian extensions in terms of Artin map, rudimentary class field theory, Examples - quadratic and cyclotomic extensions.<br /> 5. Arithmetic of cyclotomic fields, the Kronecker-Weber Theorem.<br /> 6. Dedekind zeta function, Kronecker's limit formula giving the residue of the Dedekind zeta function of a number field at s=1 in terms of its class number, regulator and discriminant.<br /> <br /> In addition, time permitting, the instructor may want to add to the list topics of special to him/her.<br /> <br /> <br /> <div style="-moz-column-count:2; column-count:2;"><br /> <br /> </div><br /> === Textbooks ===<br /> <br /> Possible textbooks for this course include (but are not limited to):<br /> <br /> *<br /> <br /> === Additional topics ===<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|686]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_686R:_Topics_in_Algebraic_Number_Theory.&diff=1582 Math 686R: Topics in Algebraic Number Theory. 2010-10-29T01:10:29Z

<p>Jsc3: </p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Topics in Algebraic Number Theory.<br /> <br /> === Credit Hours ===<br /> 3<br /> <br /> === Prerequisite ===<br /> [[Math 352, 487, 671, 672]].<br /> <br /> === Description ===<br /> Current topics of research interest.<br /> <br /> == Desired Learning Outcomes ==<br /> To gain familiarity with working in general settings, e.g. Abelian varieties over number fields, not just of elliptic curves over rationals.<br /> === Prerequisites ===<br /> A sound understanding of basic concepts of complex analysis, abstract algebra and number theory, at the level of Math 352, 371, 372 and 487 may suffice.<br /> === Minimal learning outcomes ===<br /> <br /> 1. Generalization of the Unique Factorization Theorem from rationals to number fields: Basic Definitions - number fields, algebraic integers in a number field, integral bases, discriminant, norms of ideals, finiteness of ideals of bounded norm, class number, finiteness of class number, Dedekind's Unique Factorization Theorem for ideals of a number field.//<br /> 2. Geometry of numbers: Minkowski's lemma on lattice points, Logarithmic spaces, Dirichlet's Unit Theorem for the units of the ring of integers of a number field, theorems of Minkowski and of Hermite on discriminants of number fields.//<br /> 3. Ramification Theory: Relative extensions, relative discriminant and Dedekind's criterion for ramification in terms of discriminant, higher ramification groups and Hilbert theory of ramification.//<br /> 4 Splitting of Primes: Frobenius map, Artin symbol, Artin map and splitting of primes in Abelian extensions in terms of Artin map, rudimentary class field theory, Examples - quadratic and cyclotomic extensions.<br /> 5. Arithmetic of cyclotomic fields, the Kronecker-Weber Theorem.//<br /> 6. Dedekind zeta function, Kronecker's limit formula giving the residue of the Dedekind zeta function of a number field at s=1 in terms of its class number, regulator and discriminant.//<br /> <br /> In addition, time permitting, the instructor may want to add to the list topics of special to him/her.<br /> <br /> <br /> <div style="-moz-column-count:2; column-count:2;"><br /> <br /> </div><br /> === Textbooks ===<br /> <br /> Possible textbooks for this course include (but are not limited to):<br /> <br /> *<br /> <br /> === Additional topics ===<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|686]]</div>

Jsc3 https://math.byu.edu/wiki/index.php?title=Math_686R:_Topics_in_Algebraic_Number_Theory.&diff=1581 Math 686R: Topics in Algebraic Number Theory. 2010-10-29T00:58:55Z

<p>Jsc3: </p> <hr /> <div>== Catalog Information ==<br /> <br /> === Title ===<br /> Topics in Algebraic Number Theory.<br /> <br /> === Credit Hours ===<br /> 3<br /> <br /> === Prerequisite ===<br /> [[Math 352, 487, 671, 672]].<br /> <br /> === Description ===<br /> Current topics of research interest.<br /> <br /> == Desired Learning Outcomes ==<br /> Students should gain a familiarity with working in general settings, e.g. Abelian varieties over number fields, instead of elliptic curves over rationals.<br /> === Prerequisites ===<br /> A sound understanding of basic concepts of complex analysis, abstract algebra and number theory, at the level of Math 352, 371, 372 and 487 may suffice.<br /> === Minimal learning outcomes ===<br /> <br /> 1. Generalization of the Unique Factorization Theorem from rationals to number fields: Basic Definitions - number fields, algebraic integers in a number field, integral bases, discriminant, norms of ideals, finiteness of ideals of bounded norm, class number, finiteness of class number, Dedekind's Unique Factorization Theorem for ideals of a number field.<br /> 2. Geometry of numbers: Minkowski's lemma on lattice points, Logarithmic spaces, Dirichlet's Unit Theorem for the units of the ring of integers of a number field, theorems of Minkowski and of Hermite on discriminants of number fields.<br /> 3. Ramification Theory: Relative extensions, relative discriminant and Dedekind's criterion for ramification in terms of discriminant, higher ramification groups and Hilbert theory of ramification.<br /> 4 Splitting of Primes: Frobenius map, Artin symbol, Artin map and splitting of primes in Abelian extensions in terms of Artin map, rudimentary class field theory, Examples - quadratic and cyclotomic extensions.<br /> 5. Arithmetic of cyclotomic fields, the Kronecker-Weber Theorem.<br /> 6. Dedekind zeta function, Kronecker's limit formula giving the residue of the Dedekind zeta function of a number field at s=1 in terms of its class number, regulator and discriminant.<br /> <br /> In addition, time permitting, the instructor may want to add to the list topics of special to him/her.<br /> <br /> <br /> <div style="-moz-column-count:2; column-count:2;"><br /> <br /> </div><br /> === Textbooks ===<br /> <br /> Possible textbooks for this course include (but are not limited to):<br /> <br /> *<br /> <br /> === Additional topics ===<br /> <br /> === Courses for which this course is prerequisite ===<br /> <br /> [[Category:Courses|686]]</div>

Jsc3