Idealklassen und Einheiten in kubischen Körpern

Scholz, Arnold

doi:10.1007/bf01708865

Cited by 20 publications

(15 citation statements)

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“…The theory of second p-class groups was initiated by Scholz and Taussky [38,39], using Schreier's concept of group extensions [36,37] Eick, et al [11,10,12].…”

Section: Introductionmentioning

confidence: 99%

“…In section 4 we use the theory of dihedral field extensions, in particular some well-known class number relations by Scholz and Moser [38,31], to show that in the case of a quadratic base field K = Q( √ D) and an odd prime p ≥ 3 the invariants of the second p-class group G determine the p-class numbers h p (N i ) of the dihedral fields N i and the p-class numbers h p (L i ) of the non-Galois subfields L i of absolute degree p of the N i . In contrast to [25], where we have solved the multiplicity problem for discriminants of dihedral fields which are ramified with conductor f > 1 over their quadratic subfield, we are now concerned exclusively with unramified extensions having f = 1.…”

Section: Introductionmentioning

confidence: 99%

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THE SECOND p-CLASS GROUP OF A NUMBER FIELD

Mayer¹

2012

Int. J. Number Theory

115

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Abstract. For a prime p ≥ 2 and a number field K with p-class group of type (p, p) it is shown that the class, coclass, and further invariants of the metabelian Galois group G = Gal(F 2 p (K)|K) of the second Hilbert p-class field F 2 p (K) of K are determined by the p-class numbers of the unramified cyclic extensions N i |K, 1 ≤ i ≤ p + 1, of relative degree p. In the case of a quadratic field K = Q( √ D) and an odd prime p ≥ 3, the invariants of G are derived from the p-class numbers of the non-Galois subfields L i |Q of absolute degree p of the dihedral fields N i . As an application, the structure of the automorphism group G = Gal(F 2 3 (K)|K) of the second Hilbert 3-class field F 2 3 (K) is analysed for all quadratic fields K with discriminant −10 6 < D < 10 7 and 3-class group of type (3, 3) by computing their principalisation types. The distribution of these metabelian 3-groups G on the coclass graphs G(3, r), 1 ≤ r ≤ 6, in the sense of Eick and Leedham-Green is investigated. IntroductionFor an algebraic number field K and a prime p ≥ 2 we denote by Cl p (K) the p-class group of K, that is the Sylow p-subgroup Syl p Cl(K) of its ideal class group. In this paper we shall be concerned with number fields having an elementary abelian bicyclic p-class group of type (p, p). We define the p-class field tower of The aim of the present paper is to investigate the second p-class group G = Gal(F 2 p (K)|K), that is the Galois group of the two-stage tower,, of a base field K with p-class group of type (p, p). The second p-class group G is distinguished by the special property that its commutator subgroupis isomorphic to the p-class group of the first Hilbert p-class field of K, whence G is metabelian.The theory of second p-class groups was initiated by Scholz and Taussky [38,39], using Schreier's concept of group extensions [36,37] Eick, et al. [11,10,12].

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“…The theory of second p-class groups was initiated by Scholz and Taussky [38,39], using Schreier's concept of group extensions [36,37] Eick, et al [11,10,12].…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

THE SECOND p-CLASS GROUP OF A NUMBER FIELD

Mayer¹

2012

Int. J. Number Theory

115

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“…This result was proved analytically and held without the restrictions that K 2 be complex and that K 6 be unramified over K 2 . In 1933, Scholz [13] showed that if one makes the above restrictions then £ is equal to 1. However neither a purely algebraic proof of (10) nor a statement of (9) appears to have been published.…”

Section: K = I|3'|amentioning

confidence: 99%

“…These two papers provide a general outline of the structure of the field K 6 and its subfields. Using the techniques of Furtwangler's famous paper [7], Scholz [13] was able, in 1933, to establish the second part of Theorem 4.2 and later, together with Taussky [14], to analyse the case Cl 2 = C(3) x C(3), where C(«) denotes the cyclic group of order n. Recently, Barrucand and Cohn [2], [3] and Honda [11] have examined pure cubic fields, i.e. Q(n l/3 ) for some integer n. In a different direction Shanks [15], [16] and Craig [5] have established the existence of quadratic fields with r 2 = 3 and r 2 = 4.…”

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confidence: 99%

The 3‐class groups of non‐Galois cubic fields—I

Callahan

1974

Mathematika

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“…Finally, as an application of the notions of multiplets and DPF types, the Scholz conjecture [39] concerning the distinguished index of subfield units (U N : U 0 ) = 1 of the normal closure N of L is stated, refined, and proved completely in Section 9.…”

Section: Introductionmentioning

confidence: 99%

Classifying multiplets of totally real cubic fields

Mayer¹

2021

Electronic Journal of Mathematics

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The number of non-isomorphic cubic fields L sharing a common discriminant dL = d is called the multiplicity m = m(d) of d. For an assigned value of d, these fields are collected in a homogeneous multiplet M d = (L1, . . . , Lm). By entirely new techniques for the construction and classification, we determine the differential principal factorization types τ (Li) ∈ {α1, α2, α3, β1, β2, γ, δ1, δ2, ε} of the members Li of each multiplet M d of non-cyclic totally real cubic fields with discriminants d < 10 7 . This is a new kind of arithmetical invariants which provide succinct information about ambiguous principal ideals and capitulation in the normal closures N of non-Galois cubic fields L. The classification is arranged with respect to increasing 3-class rank of the quadratic subfields K of the S3-fields N , and to ascending number of prime divisors of the conductor f of N/K. The Scholz conjecture concerning the distinguished index of subfield units (UN : U0) = 1 for ramified extensions N/K with conductor f > 1 is also verified by a tremendous minimal discriminant d = 18 251 060 outside of the systematic range d < 10 7 .

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Idealklassen und Einheiten in kubischen Körpern

Cited by 20 publications

References 3 publications

THE SECOND p-CLASS GROUP OF A NUMBER FIELD

THE SECOND p-CLASS GROUP OF A NUMBER FIELD

The 3‐class groups of non‐Galois cubic fields—I

Classifying multiplets of totally real cubic fields

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