Galois action on class groups

Lemmermeyer, Franz

doi:10.1016/s0021-8693(03)00122-4

Cited by 7 publications

(3 citation statements)

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“…The first special case of this proposition was found by Kummer, who used a similar technique for showing that the class group of Q(ζ 29 ) has type (2, 2, 2). For proofs of more general results in this direction see [5]; the essential lines in the historical development were described by Metsänkylä [6].…”

Section: Galois Actionmentioning

confidence: 99%

Grün’s theorems and class groups

Lemmermeyer¹

2010

Arch. Math.

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Abstract. In this article we show how Grün's results in group theory can be used for studying the structure of class groups in normal extensions.Otto Grün, who is known for his contributions to group theory, was an amateur mathematician; for details on his "remarkable career" and his correspondence with Hasse, see Roquette's article [8]. Grün's idea of studying the action of the Galois group on class groups led him into problems in representation theory and group theory. In this article, I would like to explain how Grün's results can be applied to class field theory.

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Section: Galois Actionmentioning

confidence: 99%

Grün’s theorems and class groups

Lemmermeyer¹

2010

Arch. Math.

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“…Consider a cyclic extension F/Q of prime degree p, and assume that 2 is a primitive root modulo p. Since the cyclic group G = Gal (F/Q) acts on class groups and units groups, we find (see e.g. [20]) that the dimensions of the 2-class groups Cl 2 (F ) and Cl Now assume that h + > h, where h and h + denote the class numbers of F in the usual and the strict sense. In this case, dim…”

Section: Associated Unit Groupsmentioning

confidence: 99%

“…Consider a cyclic extension F/Q of prime degree p, and assume that 2 is a primitive root modulo p. Since the cyclic group G = Gal (F/Q) acts on class groups and units groups, we find (see e.g. [21]) that the dimensions of the 2-class groups Cl 2 (F ) and Cl + 2 (F ), as well as of E + /E 2 and E + 4 /E 2 (note that G acts fixed point free on E + /E 2 , but not on E/E 2 ) as F 2 -vector spaces are all divisible by p − 1. Since ρ + − ρ ≤ p−1 2 by Armitage-Fröhlich, we conclude that ρ + = ρ.…”

Section: 2mentioning

confidence: 99%

Selmer groups and quadratic reciprocity

Lemmermeyer

2006

Abh.Math.Semin.Univ.Hambg.

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Abstract. In this article we study the 2-Selmer groups of number fields F as well as some related groups, and present connections to the quadratic reciprocity law in F .Let F be a number field; elements in F × that are ideal squares were called singular numbers in the classical literature. They were studied in connection with explicit reciprocity laws, the construction of class fields, or the solution of embedding problems by mathematicians like Kummer, Hilbert, Furtwängler, Hecke, Takagi, Shafarevich and many others. Recently, the groups of singular numbers in F were christened Selmer groups by H. Cohen [4] because of an analogy with the Selmer groups in the theory of elliptic curves (look at the exact sequence (2.2) and recall that, under the analogy between number fields and elliptic curves, units correspond to rational points, and class groups to Tate-Shafarevich groups).In this article we will present the theory of 2-Selmer groups in modern language, and give direct proofs based on class field theory. Most of the results given here can be found in § § 61ff of Hecke's book [11]; they had been obtained by Hilbert and Furtwängler in the roundabout way typical for early class field theory, and were used for proving explicit reciprocity laws. Hecke, on the other hand, first proved (a large part of) the quadratic reciprocity law in number fields using his generalized Gauss sums (see [3] and [22]), and then derived the existence of quadratic class fields (which essentially is just the calculation of the order of a certain Selmer group) from the reciprocity law.In Takagi's class field theory, Selmer groups were moved to the back bench and only resurfaced in his proof of the reciprocity law. Once Artin had found his general reciprocity law, Selmer groups were history, and it seems that there is no coherent account of their theory based on modern class field theory.Hecke's book [11] is hailed as a classic, and it deserves the praise. Its main claim to fame should actually have been Chapter VIII on the quadratic reciprocity law in number field, where he uses Gauss sums to prove the reciprocity law, then derives the existence of 2-class fields, and finally proves his famous theorem that the ideal class of the discriminant of an extension is always a square. Unfortunately, this chapter is not exactly bedtime reading, so in addition to presenting Hecke's results in a modern language I will also give exact references to the corresponding theorems in Hecke's book [11] in the hope of making this chapter more accessible.The actual reason for writing this article, however, was that the results on Selmer groups presented here will be needed for computing the separant class group of F , a new invariant that will be discussed thoroughly in [21].

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Invariant generalized ideal classes – structure theorems for p-class groups in p-extensions

Gras¹

2017

Proc Math Sci

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We give, in Sections 2 and 3, an english translation of: Classes généralisées invariantes, J. Math. Soc. Japan, 46, 3 (1994), with some improvements and with notations and definitions in accordance with our book: Class Field Theory: from theory to practice, SMM, Springer-Verlag, 2 nd corrected printing 2005. We recall, in Section 4, some structure theorems for finite Zp[G]-modules (G ≃ Z/p Z) obtained in: Sur les ℓ-classes d'idéaux dans les extensions cycliques relatives de degré premier ℓ, Annales de l 'Institut Fourier, 23, 3 (1973). Then we recall the algorithm of local normic computations which allows to obtain the order and (potentially) the structure of a p-class group in a cyclic extension of degree p. In Section 5, we apply this to the study of the structure of relative p-class groups of Abelian extensions of prime to p degree, using the Thaine-Ribet-Mazur-Wiles-Kolyvagin "principal theorem", and the notion of "admissible sets of prime numbers" in a cyclic extension of degree p, from: Sur la structure des groupes de classes relatives, Annales de l'Institut Fourier, 43, 1 (1993). In conclusion, we suggest the study, in the same spirit, of some deep invariants attached to the p-ramification theory (as dual form of non-ramification theory) and which have become standard in a p-adic framework. Since some of these techniques have often been rediscovered, we give a substantial (but certainly incomplete) bibliography which may be used to have a broad view on the subject.We shall see the field L given, via Class Field Theory, by some Artin group of K (e.g., the Hilbert class field H + K of K associated with the group of principal ideals, in the narrow sense, any ray class field H + K,m associated with a ray group modulo a modulus m of k, in the narrow sense, or more generally any subfield L of these canonical fields, defining Gal(H + K,m /L) by means of a sub-G-module H of the generalized class group Cℓ + K,m ≃ Gal(H + K,m /K)). We intend to give, from the arithmetic of k and elementary local normic computations in K/k, an explicit formula for #Gal(L/K) G = #(Cℓ + K,m /H) G .This order is the degree, over K, of the maximal subfield of L (denoted L ab ) which is Abelian over k.

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Galois action on class groups

Cited by 7 publications

References 28 publications

Grün’s theorems and class groups

Grün’s theorems and class groups

Selmer groups and quadratic reciprocity

Invariant generalized ideal classes – structure theorems for p-class groups in p-extensions

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