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Construction of All Cubic Fields of a Fixed Fundamental Discriminant (Renate Scheidler)

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Construction of All Cubic Fields of a Fixed Fundamental Discriminant (Renate Scheidler) Chapter 4 Construction of All Cubic Fields of a Fixed Fundamental Discriminant (Renate Scheidler) 4.1 Introduction In 1925, Berwick [19] described an approach for generating all cubic fields of a given discriminant Δ. When Δ is fundamental, Berwick’s observation, expressed in modern terminology, was that every cubic field of discriminant√ Δ arises from a 3-virtual unit in the quadratic resolvent field L = Q( Δ ) of (also fundamental) discriminant Δ = −3Δ/gcd(3,Δ)2. A 3-virtual unit of L is defined to be a gen- erator of a principal ideal that√ is the cube of some ideal in the maximal order OL of L . Suppose λ =(G + H Δ )/2, with G√,H ∈ Z non-zero, is a 3-virtual unit that is not itself a cube in L. Put λ =(G − H Δ )/2 and λλ = A3 with A ∈ Z. Then f (x)=x3 − 3Ax + G is the generating polynomial of a cubic field of discriminant Δ or −27Δ , and every cubic field of discriminant Δ arises in this way. Moreover, two 3-virtual units λ1,λ2 ∈ OL give rise to the same cubic field up to Q-isomorphism if and only if λ1/λ2 or λ1/λ 2 is a cube in L . In this case, if λi is a generator of O 3 α the L -ideal ai for i = 1,2, then a1 is equivalent to a2 or to a2, i.e., a1 =( )a2 or a1 =(α)a2 for some non-zero α ∈ L . Cubic fields of fundamental discriminant Δ can therefore be obtained from 3-virtual units in the quadratic resolvent field of discriminant Δ , or more exactly, via the cube roots of ideals belonging to 3-torsion classes in the class group of L. Care must be taken that this construction produces the complete collection of triples of conjugate cubic fields of discriminant Δ, that it yields each such field exactly once, and that any fields of discriminant −27Δ are detected and eliminated. A major problem with Berwick’s approach is that the generating polynomials thus obtained can have extremely large coefficients, particularly when Δ < 0, in which case L is a real quadratic field. An ingenious solution to this problem was devised by Shanks who proposed a 3-virtual unit construction that produces generat- ing polynomials with remarkably small coefficients. For example, for the 13 triples © Springer Nature Switzerland AG 2018 173 S. A. Hambleton, H. C. Williams, Cubic Fields with Geometry, CMS Books in Mathematics, https://doi.org/10.1007/978-3-030-01404-9 4 174 4 Construction of All Cubic Fields of a Fixed Fundamental Discriminant (Renate Scheidler) of conjugate cubic fields of discriminant Δ = 44806173, Shanks’ algorithm pro- duces the following generating polynomials: 3 2 f1(x)=x − 61x + 697x − 330, 3 2 f2(x)=x − 279x + 441x − 170, 3 2 f3(x)=x − 63x + 423x − 8, 3 2 f4(x)=x − 69x + 435x − 216, 3 2 f5(x)=x − 63x + 603x − 494, 3 2 f6(x)=x − 83x + 297x − 54, 3 2 f7(x)=x − 63x + 837x − 494, (4.1) 3 2 f8(x)=x − 257x + 477x − 216, 3 2 f9(x)=x − 87x + 273x − 36, 3 2 f10(x)=x − 62x + 546x − 261, 3 2 f11(x)=x − 60x + 660x − 97, 3 2 f12(x)=x − 165x + 273x − 90, 3 2 f13(x)=x − 127x + 185x − 62. In particular, in the computationally more interesting case of negative cubic dis- criminants, all computations are conducted in the set of reduced ideals in each 3-torsion ideal class of the class group of L, known as the infrastructure of the class. Shanks therefore assigned his algorithm the six-letter FORTRAN designator CUFFQI, pronounced “cuff-key,” an acronym derived from the phrase Cubic Fields From Quadratic Infrastructure. Shanks’ work is described in his talks [170, 172] and in a manuscript [171] dating back to 1987, but was never published. In his 1990 doctoral dissertation, Fung [77] presented CUFFQI in a computa- tionally more suitable form and implemented it in FORTRAN on an Amdahl 5870 mainframe computer. Evidence of the efficiency of Fung’s version of CUFFQI is provided by his impressive (for the late 1980s) computation of all 364 non-conjugate cubic fields of the 19-digit discriminant Δ = −3161659186633662283 in under 3 CPU minutes. This chapter provides a modern description of the previously unpub- lished Shanks-Fung CUFFQI algorithm for constructing all cubic fields of a given fundamental discriminant Δ. 4.2 The Quadratic Fields Associated with a Cubic Field Let K be a cubic field of discriminant√ Δ. Recall from §1.6 that there are two fields√ as- sociated with K, namely L = Q( Δ) and its dual (or mirror) field L = Q( −3Δ) 4.2 The Quadratic Fields Associated with a Cubic Field 175 which is the resolvent field of K.IfD is the (fundamental) discriminant of L, related to Δ via (1.68), then the discriminant of L, termed the dual discriminant, is −3D −3D if 3 D, D = = (4.2) gcd(D,3)2 −D/3if3| D, Note that {D,D} = {1,−3} if and only if one of L, L is Q; otherwise, both L and L are quadratic extensions of Q. If Δ is a fundamental discriminant, then D = Δ, and the number of cubic fields of discriminant D is related to L as follows [95, Satz 7, p. 587]: Theorem 4.1. Let L be a quadratic field of discriminant D, and r the 3-rank of the class group of L. Then the number of non-conjugate cubic fields of discriminant D is (3r − 1)/2. √ For example, the class group of the real quadratic field L = Q( 44806173) has 3- rank 3, so there are (33 −1)/2 = 13 non-conjugate cubic fields of that discriminant, generated by the 13 polynomials√ listed in (4.1). Quer [156] determined that the class group of L = Q( −3161659186633662283) has 3-rank 6, so there are 364 non-conjugate cubic fields with this discriminant, for which Fung found generating polynomials in [77]. The 3-ranks of the ideal class groups of L and its dual field L are closely related through a theorem due to [165]: Theorem 4.2. Let D and D be dual fundamental discriminants with D < 0, and let r and r denote the respective√ 3-ranks of the ideal class groups√ of the imaginary quadratic field L = Q D) and the real quadratic field L = Q( D). Then r = r or r = r + 1. The first of these two cases is referred to as non-escalatory, whereas√ the second case is labelled escalatory√ [173]. For example, the field L = Q( −14935391) and its dual L = Q( 44806173) both have class groups of 3-rank 3 and hence belong to the non-escalatory case. Larger examples include the escalatory quadratic field √ L = Q( −35102371403731) of 3-rank 5 and the non-escalatory field √ L = Q( −250930267537731) of 3-rank 4; see Section 5.9 of [77]. More recently, Kishi [111] characterized the escalatory scenario by linking it to the existence of cubic fields with certain prop- erties and to solutions of norm equations in L. Among other criteria, he proved that r = r + 1 if and only if there does not exist a triple (x,y,z) ∈ Z3 such that gcd(x,y)=1, x2 ≡ 1or7 (mod 9), y ≡ 1 (mod 3), z = 0, and x2 −3z2d = 4y3.Here − − ≡ − D d > 0 is the square-free part of D, i.e., d = D if D 1 (mod 4) and d = 4 otherwise. 176 4 Construction of All Cubic Fields of a Fixed Fundamental Discriminant (Renate Scheidler) The resolvent field L is further related to the cubic field K through the roots of a generating polynomial. As seen in §1.4, there exists a generating polynomial of K of the form f (x)=x3 − 3Ax + G (4.3) 2 3 with A,G ∈ Z. The discriminant of f (x) is D f = 27(4A − G ). Recall from (1.52) that the roots βi, i = 0,1,2, of f (x) are given by √ √ i 3 −i 3 βi = η μ + η ν, i = 0,1,2, where η is a primitive cube root of unity and μ,ν are given by (1.51). Put √ G + G2 − 4A3 λ = −ν = . (4.4) 2 Then the minimal polynomial of λ over Q is R(x)=x2 − Gx + A3 of discriminant 2 3 DR = G − 4A = −D f /27. Since DR/D is a square, it follows that λ ∈ L , and we have A β = −ηiκ − with κ3 = λ, i = 0,1,2. (4.5) i ηiκ In this way, every cubic field K, through a generating polynomial of the form (4.3), defines an element λ in the maximal order OL of the resolvent field L of K.Fol- lowing the terminology of Berwick [19], we call λ a (quadratic) generator of K. Note that both λ and λ are quadratic generators of each of the three conjugate cubic fields Q(βi), i = 0,1,2. We will make use of the following useful auxiliary result. Recall that the Galois closure of K is obtained by adjoining any of the three roots βi, i = 0,1,2, to L.The analogous construction over L yields three different cubic extensions of L, and we have the following field equalities: i Lemma 4.1. Let κ and βi (i = 0,1,2) be given by (4.5). Then L (βi)=L (η κ). i i Proof. By (4.5), βi ∈ Q(η κ) ⊂ L (η κ).
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