A EUCLIDEAN RING CONTAINING ℤ[\sqrt{14}]

DAVID A. CARDON

Abstract. English

We show that the ring ℤ[\sqrt{14}, 1/2] is Euclidean and that this ring admits a totally multiplicative Euclidean function. This example is interesting because it is related to the ring ℤ[\sqrt{14}] which is conjectured to be Euclidean but for which a proof is lacking.

French

Nous montrons que l’anneau ℤ[\sqrt{14}, 1/2] est euclidien et que cet anneau admet une fonction euclidienne totalement multiplicative. Cet exemple est intéressant car il est relié à l’anneau ℤ[\sqrt{14}], qui est conjecturé être euclidien.

Date: May 7, 1997. 1991 Mathematics Subject Classification. 11R04. Key words and phrases. Euclidean rings, algebraic number theory. Research partially supported by NSERC.

Considerable effort has been spent in trying to classify the Euclidean number fields (see Lemmermeyer [8]). An integral domain R is called Euclidean if there is a function ϕ : R → {0, 1, 2, … } such that ϕ(a) = 0 if and only if a = 0 and for every a, b ∈ R, b ≠ 0, there exist q, r ∈ R such that a = bq + r with ϕ(r) < ϕ(b). A number field is said to be Euclidean when its ring of integers is Euclidean.

Of the finite number of known examples of Euclidean number fields many are norm-Euclidean; that is, ϕ is the absolute value of the norm. It is well known that there are only finitely many norm-Euclidean real quadratic number fields. However, Weinberger [11] proved that if one assumes an appropriate Riemann Hypothesis then a ring of algebraic integers which is a principal ideal domain and has infinitely many units is Euclidean. Lenstra [9] extended Weinberger’s result to rings of S-integers and gave an explicit Euclidean function. Clark [3] found the first example of a real quadratic number field that is Euclidean but not for the norm, the field being ℚ(\sqrt{69}). Weinberger’s result predicts that ℚ(\sqrt{14}) as well as many other number fields should also be Euclidean.

Without using a Riemann hypothesis Samuel [10] and Bedocchi [2] provide evidence that suggests ℤ[\sqrt{14}] might be Euclidean. In this paper we will show that the related ring ℤ[\sqrt{14}, 1/2] is Euclidean with a completely multiplicative Euclidean function. Harper [5], by a more complicated method than that presented here, has shown that ℤ[\sqrt{14}, 1/p] is Euclidean for every prime p. However, the Euclidean functions constructed by that argument are not completely multiplicative.

Proposition 1. Let η = 4 + \sqrt{14}. (This is the prime above 2 in the ring ℤ[\sqrt{14}].) The ring ℤ[\sqrt{14}, 1/2] is Euclidean with a completely multiplicative Euclidean function ϕ given by

ϕ(x) = | Norm(x′)|

where Norm(a + b\sqrt{14}) = a^{2} − 14b^{2} denotes the norm of an element in ℚ(\sqrt{14}) and x is written as x = x′η^{α} such that x′ contains no power of η.

The proposition says that if 2 is made into a unit then the ring ℤ[\sqrt{14}] becomes the Euclidean ring ℤ[\sqrt{14}, 1/2]. The Euclidean function given above is the absolute norm in the ring R = ℤ[\sqrt{14}, 1/2]. Thus, ϕ(x) = |R/(x)| which is the number of number of cosets modulo the principal ideal (x). This is also called the S-norm on ℚ(\sqrt{14}) where S is the set consisting of the archimedean primes and the prime η = 4+\sqrt{14}. The proof of proposition 1 will follows from lemmas 2 and 3 below.

Each element α + β\sqrt{14} ∈ ℚ(\sqrt{14}) may be thought of as a point (α, β) in the plane ℝ^{2}. We will use a geometrical argument for the quadratic form F (x, y) = x^{2} − 14y^{2} which represents the norm of x + y\sqrt{14}. Let Λ be the lattice of points with integer coordinates in ℝ^{2}. Every point in ℝ^{2}/Λ may be represented by a point in the fundamental domain

D_{Λ} = {(x, y) | 0 ≤ x < 1 and 0 ≤ y < 1}.

Let T be the linear transformation T on ℝ^{2} given by

(x′, y′) = T (x, y) = (15x + 56y, 4x + 15y).

Then F (x′, y′) = F (x, y). The transformation T is referred to as an automorph in [1]. The powers of T are related to the solutions of the Pellian equation x^{2}−14y^{2} = 1. For a given lattice L in ℝ^{2} define the inhomogeneous minimum of a point (x, y) ∈ ℝ^{2} to be

M_{L}(x, y) = lim inf_{(a,b)∈L} |F (x + a, y + b)|.

Then it is apparent (as explained in [1]) that M_{L}(x, y) = M_{L}(T (x, y)). If it were the case that M_{Λ}(x, y) < 1 for every (x, y) ∈ ℚ^{2} we could deduce that ℤ[\sqrt{14}] was norm-Euclidean. Although this is not the case, the following lemma shows it is almost true.

Lemma 2. M_{Λ}(x, y) < 1 unless (x, y) ≡ (1/2, 1/2) mod Λ. In the exceptional case, M_{Λ}(1/2, 1/2) = 5/4.

Remark. Lemmas 2 and 3 are given as Propositions 3.1 and 3.2 in [2]; however, the proof given here is simpler.

Proof of Lemma 2 . It is sufficient to prove the lemma for points in D_{Λ}. We first show the lemma is true for points not too close to (1/2, 1/2).

Consider the vectors ē_{1} = (\sqrt{14/15}, 1/\sqrt{15}) and ē_{2} = (\sqrt{14/15}, −1/\sqrt{15}). Then ē_{1} and ē_{2} are normalized eigenvectors of T such that T (ē_{1}) = uē_{1} and T (ē_{2}) = u^{−1}ē_{1} where u = 15 + 4\sqrt{14}. Let ϵ = \frac{1}{2u} and let P be the diamond shaped region

P = {(x, y) ∈ ℝ^{2} | (x, y) = (1/2, 1/2) + αē_{1} + βē_{2}, |α| < ϵ, |β| < ϵ}.

We divide the square D_{Λ} into very small squares and check by computer that each of the small square has a translate modulo Λ lying strictly between the hyperbolas F (x, y) = ±1 unless the small square is entirely contained in the region P.

To deal with the exceptional points lying inside P we note that (1/2, 1/2) is a fixed point of T modulo Λ. If (x, y) = (1/2, 1/2) + αē_{1} + βē_{2} ∈ P and α ≠ 0, choose the smallest k > 0 such that |αu^{k}| > ϵ but |αu^{k−1}| < ϵ. Then T^{k}(x, y) ∈ D_{Λ} mod Λ, but T^{k}(x, y) ∉ P mod Λ. This shows that (x, y) has a representative (x̃, ỹ) with F (x̃, ỹ) < 1 so that M_{Λ}(x, y) < 1. If β ≠ 0 we argue similarly using negative powers of T.

The value M_{Λ}(1/2, 1/2) = 5/4 dates back to Heinhold [7].

The following modification of the previous lemma applies to arithmetic in ℤ[\sqrt{14}] modulo an ideal.

Lemma 3. Let a, b, c, d ∈ ℤ.

1. Suppose c and d are not both even. Then there is a representative ã + b̃\sqrt{14} of a + b\sqrt{14} modulo c + d\sqrt{14} such that

N(ã + b̃\sqrt{14}) < N(c + d\sqrt{14}).

2. If c and d are both even, the first part of the lemma holds except that the smallest representative of (\frac{c}{2} + \frac{d}{2} \sqrt{14})(1 + \sqrt{14}) modulo c + d\sqrt{14} has absolute norm \frac{5}{4}N(c + d\sqrt{14}).

Proof. Let Λ be the lattice of points with generators (1, 0) and (0, 1). Any point of ℝ^{2} modulo Λ may be represented in a fundamental domain

D_{Λ} = {(s, t) | 0 ≤ s < 1 and 0 ≤ t < 1}.

Similarly, let Γ be the lattice with generators (c, d) and (14d, c) and fundamental domain

D_{Γ} = {s(c, d) + t(14d, c) | 0 ≤ s < 1 and 0 ≤ t < 1}.

Then Λ represents the whole ring ℤ[\sqrt{14}], but Γ represents the ideal generated by c + d\sqrt{14}. To adapt lemma 2 which applies to the case of arithmetic modulo Λ to the case of arithmetic modulo Γ we map (x, y) ∈ D_{Λ} to (x′, y′) ∈ D_{Γ} by the linear transformation

(x′, y′) = (cx + 14dy, dx + cy).

Then the quadratic form F (x, y) = x^{2} − 14y^{2} at (x′, y′) becomes

F (x′, y′) = (c^{2} − 14d^{2})(x^{2} − 14y^{2}) = (c^{2} − 14d^{2})F (x, y).

Thus, the linear transformation has simply rescaled F (x, y). The ‘exceptional’ point (1/2, 1/2) ∈ D_{Λ} maps to the new ‘exceptional’ point (c/2 + 7d, d/2 + c/2) ∈ D_{Γ}. Hence, if (1/2, 1/2) ≠ (x, y) ∈ D_{Λ}, then

M_{Γ}(x′, y′) = |c^{2} − 14d^{2}| M_{Λ}(x, y) < |c^{2} − 14d^{2}|

whereas

M_{Γ}(c/2 + 7d, d/2 + c/2) = |c^{2} − 14d^{2}| M_{Λ}(1/2, 1/2) = \frac{5}{4}|c^{2} − 14d^{2}|.

This proves the lemma.

Proposition 1 is an immediate consequence of lemma 3. If 1/2 is adjoined to ℤ[\sqrt{14}] then the elements 2 and η = 4+\sqrt{14} become units in ℤ[\sqrt{14}, 1/2]. The ordinary prime 2 ramifies in ℤ[\sqrt{14}] and the prime above it is η. Define a completely multiplicative function ϕ on ℤ[\sqrt{14}, 1/2] by

ϕ(x) = | Norm(x′)|

where Norm(a + b\sqrt{14}) = a^{2} − 14b^{2} denotes the norm of an element in ℚ(\sqrt{14}) and x is written as x = x′η^{α} such that x′ contains no power of η. By inverting 2 we eliminate the difficulty caused by the evenness of c and d in the second part of lemma 3. Thus given any a, b ∈ ℤ[\sqrt{14}, 1/2] with b ≠ 0 there exist q and r such that a = bq + r and ϕ(r) < ϕ(b). This completes the proof of proposition 1.

Acknowledgment. I would like to thank David Clark for introducing me to the problem of Euclidean number fields during a brief visit to Brigham Young University in 1995, and I wish to thank Ram Murty of Queen’s University for several useful conversations. The referee made several suggestions which improved this paper. Also, the papers of Hartman [6] and Elsner [4] influenced my thinking as I studied this problem.

References

[1] E.S. Barnes and H.P.F. Swinnerton-Dyer, The inhomogeneous minima of binary quadratic forms (I), Acta Mathematica 87 (1952), 259-323.

[2] Edmondo Bedocchi, L’anneau ℤ[\sqrt{14}] et l’algorithme Euclidien, Manuscripta Mathematica 53 (1985), 199-216.

[3] David A. Clark, A quadratic field which is Euclidean but not norm-Euclidean, Manuscripta Mathematica 83 (1994), 327-330.

[4] C. Elsner, On the approximation of irrational numbers with rationals restricted by congruence relations, Fibonacci Quarterly, Vol. 34, No. 1, (1996), 18-29.

[5] Malcom Harper, McGill University Masters Thesis, (1997).

[6] S. Hartman, Sur une condition supplémentaire dans les approximations diophantiques, Colloquium Mathematicum, Vol. 2, (1951), 48-51.

[7] Joseph Heinhold, Verallgemeinerung und Verschärfung eines Minkowskischen Satzes, Mathematische Zeitschrift 44 (1938), 659-688.

[8] Franz Lemmermeyer, The Euclidean algorithm in algebraic number fields, Expositiones Mathematicae 13 (1995), 385-416.

[9] H.W. Lenstra On Artin’s Conjecture and Euclid’s Algorithm in Global Fields, Inventiones Mathematicae 42 (1977), 201-224.

[10] P. Samuel, About Euclidean Rings, Journal of Algebra 19 (1971), 282-301.

[11] Peter J. Weinberger, On Euclidean rings of algebraic integers, Analytic Number Theory, Proceedings of Symposia in Pure Mathematics, AMS, Volume XXIV (1972), 321-332.

Department of Mathematics and Statistics, Queen’s University, Kingston, Ontario K7L 3N6

E-mail address: cardon@mast.queensu.ca