For a real quadratic field F=Q(m)F=\mathbb{Q}(\sqrt{m}) whose ring of integers OF\mathcal{O}_F has class number 1, it is conjectured that OF\mathcal{O}_F is 2-stage Euclidean with respect to the norm: every pair of elements admits a division chain of length at most 2 ending with a norm-decreasing remainder. Cooke and Weinberger proved this in 1975, but only under the Generalized Riemann Hypothesis for Hecke LL-functions, and unconditional results cover only finite families — the largest verification (Guitart–Masdeu, 2013) handles all real quadratic class-number-one fields of discriminant below 8000. The open problem is an unconditional proof for all real quadratic fields of class number 1 in their natural generality, or an explicit counterexample (which would also refute GRH). Solving it is linked to the computation of continued fractions in these fields, since a 2-stage Euclidean structure yields an effective continued-fraction algorithm.

Background

A real quadratic field is F=Q(m)F=\mathbb{Q}(\sqrt{m}) for a squarefree integer m>1m>1; its ring of integers OF\mathcal{O}_F consists of the algebraic integers in FF (the elements satisfying a monic polynomial with integer coefficients), and the norm Nm(x)=x⋅x′\mathrm{Nm}(x)=x\cdot x' multiplies an element by its conjugate under the nontrivial automorphism of FF. A domain is Euclidean (with respect to the norm) if for all α,β≠0\alpha,\beta\neq 0 there is a quotient qq with ∣Nm(α−qβ)∣<∣Nm(β)∣|\mathrm{Nm}(\alpha-q\beta)|<|\mathrm{Nm}(\beta)|; iterating gives the classical Euclidean algorithm, and every Euclidean ring is a principal ideal domain, so its class number (the number of ideal classes, measuring failure of unique factorization of ideals) is 1. Motzkin and Cooke weakened this: a kk-stage division chain for (α,β)(\alpha,\beta) is a sequence of kk successive divisions α=q1β+r1\alpha=q_1\beta+r_1, β=q2r1+r2\beta=q_2r_1+r_2, …\dots, rk−2=qkrk−1+rkr_{k-2}=q_kr_{k-1}+r_k, and FF is 2-stage Euclidean (with respect to the norm) if every pair α,β∈OF\alpha,\beta\in\mathcal{O}_F, β≠0\beta\neq 0, admits such a chain of length k≤2k\le 2 with final remainder satisfying ∣Nm(rk)∣<∣Nm(β)∣|\mathrm{Nm}(r_k)|<|\mathrm{Nm}(\beta)|. A 2-stage Euclidean field still has class number 1, and, by a proposition of Samuel studied further by Chen–Leu, 2-stage Euclideanity even implies Euclideanity with respect to a suitable (non-norm) function; the point is that the remainder need only become small after two steps, a strictly weaker requirement than norm-Euclideanity. Guitart and Masdeu reformulate the condition geometrically: FF is 2-stage Euclidean if and only if a fundamental domain DD for the lattice OF\mathcal{O}_F in F⊗R≅R2F\otimes\mathbb{R}\cong\mathbb{R}^2 (a region whose OF\mathcal{O}_F-translates tile the plane; its closure is compact) is covered by finitely many hyperbolic regions V(q)={x: ∣Nm(x−q)∣<1/∣Nm(q2)∣}V(q)=\{x:\ |\mathrm{Nm}(x-q)|<1/|\mathrm{Nm}(q_2)|\}, bounded by hyperbolas, indexed by length-2 continued fractions q=[q1,q2]=q1+1/q2q=[q_1,q_2]=q_1+1/q_2 with coefficients q1,q2∈OFq_1,q_2\in\mathcal{O}_F. Norm-Euclidean real quadratic fields form the small classical list m∈{2,3,5,6,7,11,13,17,19,21,29,33,37,41,57,73}m\in\{2,3,5,6,7,11,13,17,19,21,29,33,37,41,57,73\} (each field on the list is unconditionally norm-Euclidean; completeness of the list is known under GRH), while many more class-number-one fields are known to be 2-stage Euclidean case by case. The conjectural statement — that class number 1 and 2-stage Euclideanity coincide for real quadratic fields — was recorded in Lemmermeyer's 1995 survey and goes back to the weakening of the Euclidean property introduced by Motzkin and developed by Cooke. Its only general proof is conditional: Cooke and Weinberger showed in 1975 that the Generalized Riemann Hypothesis (GRH) for Hecke LL-functions — the LL-series, generalizing the Riemann zeta function, attached to ideal-class characters of number fields — implies that any number ring with infinitely many units and class number 1 admits such division chains (2-stage for real quadratic fields). Removing GRH from statements of this type is possible unconditionally only in higher unit rank (the unit rank of a number field being the number of multiplicatively independent fundamental units in its ring of integers): Harper and Murty (2004) proved unconditionally that Euclidean and principal ideal domain coincide for finite Galois extensions of Q\mathbb{Q} with unit rank above 3, while real quadratic fields have unit rank 1 and are not covered. Clark (1996) removed GRH for totally real Galois extensions of degree at least 3, but for real quadratic fields his unconditional version requires an additional hypothesis — sufficiently many prime elements whose residue-class powers the unit group surjects onto — which has been verified only field by field. A further obstacle is that the quantifier 'every real quadratic field of class number 1' ranges over a family not known to be finite or infinite (it is believed infinite), so no finite computation can settle the general statement; and the algorithm of Guitart–Masdeu terminates exactly when the field is 2-stage Euclidean, so no current algorithm can certify in finite time that a field is not 2-stage Euclidean.

Problem Statement

Prove, without assuming the Generalized Riemann Hypothesis (GRH) or any other unproved hypothesis, that every real quadratic field F=Q(m)F=\mathbb{Q}(\sqrt{m}) (for squarefree m>1m>1) of class number 1 — that is, whose ring of integers OF\mathcal{O}_F has trivial ideal class group — is 2-stage Euclidean with respect to the norm Nm(x)=x⋅x′\mathrm{Nm}(x)=x\cdot x' (the product of the two real conjugates of xx): for all α,β∈OF\alpha,\beta\in\mathcal{O}_F with β≠0\beta\neq 0 there exist q1,q2,r1,r2∈OFq_1,q_2,r_1,r_2\in\mathcal{O}_F with α=q1β+r1\alpha=q_1\beta+r_1, β=q2r1+r2\beta=q_2r_1+r_2, and ∣Nm(r2)∣<∣Nm(β)∣|\mathrm{Nm}(r_2)|<|\mathrm{Nm}(\beta)|. Equivalently, that the hyperbolic regions V([q1,q2])={x∈R2: ∣Nm(x−[q1,q2])∣<1/∣Nm(q2)∣}V([q_1,q_2])=\{x\in\mathbb{R}^2:\ |\mathrm{Nm}(x-[q_1,q_2])|<1/|\mathrm{Nm}(q_2)|\}, where [q1,q2]=q1+1/q2[q_1,q_2]=q_1+1/q_2 is a length-2 continued fraction with coefficients q1,q2∈OFq_1,q_2\in\mathcal{O}_F, q2≠0q_2\neq 0, cover a fundamental domain of the lattice OF\mathcal{O}_F in R2\mathbb{R}^2 — for every such field, not merely for the finitely many fields of discriminant below 8000 already verified. Alternatively, disprove the conjecture by exhibiting a real quadratic field of class number 1 that is not 2-stage Euclidean (such a counterexample would also refute GRH for Hecke LL-functions, since the conjecture follows from GRH by Cooke–Weinberger).

References

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