Galois Extension Of Number Fields Research Articles

Taming wild extensions with Hopf algebras

Let K ⊂ L K \subset L be a Galois extension of number fields with abelian Galois group G G and rings of integers R ⊂ S R \subset S , and let A \mathcal {A} be the order of S S in K G KG . If A \mathcal {A} is a Hopf R R -algebra with operations induced from K G KG , then S S is locally isomorphic to A \mathcal {A} as A \mathcal {A} -module. Criteria are found for A \mathcal {A} to be a Hopf algebra when K = Q K = {\mathbf {Q}} or when L / K L/K is a Kummer extension of prime degree. In the latter case we also obtain a complete classification of orders over R R in L L which are tame or Galois H H -extensions, H H a Hopf order in K G KG , using a generalization of the discriminant.

Restricted ramification for imaginary quadratic number fields and a multiplicator free group

Let K be an imaginary quadratic number field with unit group EK and let t' be a rational prime such that t' IEK . Let S be any finite set of finite primes of K and let K( {, S) denote the maximal e-extension of K (inside a fixed algebraic closure of K) which is nonramified at the finite primes of K outside S. We show that the finitely generated pro-e-group Q(t, S) = Gal(K(t, S)/K) has the property that a complete set of defining relations for Q(t, S) as a pro-{-group can be obtained by lifting the nontrivial abelian or torsion relations in the maximal abelian quotient group U(t, S)ab. In addition we use the key idea of the proof to derive some interesting results on towers of fields over K with restricted ramification. Introduction. If L/K is a Galois extension of fields and E is a Galois extension of K containing L, then E is said to be a central extension of L/K if Gal(E/L) is a subgroup of the centre of Gal(E/K). The study of central extensions of Galois extensions of number fields leads to several arithmetic applications (see [4, Chapter 3]) many of which stem from an early paper of Frohlich [2]. In particular a theorem of Frohlich [4, Theorem 4.11] gives a complete description in terms of topological generators and relations of the Galois group of the maximal tl-extension of Q (t1any prime, Q the field of rational numbers) which is nilpotent of class two with ramification restricted to a finite set of primes. This, in turn, leads to another result of Frohlich [4, Theorem 5.2] which characterizes, in rational terms, all abelian e-extensions L of Q for which edoes not divide the narrow class number of L. Taking the previous discussion as motivation we see that it is important to consider the pro-e-group Q(e, S) = Gal(K(t1, S)/K), where S is a finite set of finite primes of the number field K, and K(t1, S) is the maximal tl-extension of K (inside a fixed algebraic closure of K) which is nonramified at the finite primes of K outside S. The main difficulty in describing Q(t, S) is the determination of a complete set of defining relations for Q(t, S) as a pro-tl-group. When K = Q Frohlich shows that a complete set of defining relations can be obtained by lifting the nontrivial abelian or torsion relations of the maximal abelian quotient group Q(t, S)ab. We explain this idea more precisely in ?1. Our main result shows that Q(e, S) also has the lifting property when K is an imaginary quadratic number field, S is any finite set as above and 1 + I EKI, where EK Received by the editors December 27, 1983 and, in revised form, August 17, 1984. 1980 Mathematics Subject Classification. Primary 12A65. ' This represents part of the author's Ph.D. dissertation written under the direction of Professor Stephen V. Ullom at the University of Illinois at Urbana. ?1985 American Mathematical Society 0002-9947/85 $1.00 + $.25 per page

Self-duality and torsion Galois modules in number fields

Let K/k be a Galois extension of number fields with Galois group r, and let D, o be the rings of integers of K, k, respectively. In this paper we investigate the structure of 0 as an or-module by comparing 0 with its dual, Horn@, 0). M. J. Taylor [ 11, Theorem, p. 1731 proved in 1978 that these two modules are stably isomorphic over Zr, assuming that all the primes of o dividing the order of r are unramified in K. This result had been conjectured by Frohlich [5, p. 4231 in the case k = Q. Its generalization to all tame extensions is an easy consequence of Taylor’s subsequent proof of Frohlich’s conjecture describing the class of 0 in the class group of locally free ZTmodules in terms of the Artin root number [ 12, Theorem 1, p. 4 1; 5, Proposition 3, p. 4391. To study the Galois module structure of 0 in the wild case Queyrut [7] introduced the S-Grothendieck groups Gi(oT) and K”(oT), where S is a set of primes of o. Namely, G&(oT) is the Grothendieck group corresponding to the category of finitely generated o-torsion free or-modules, with relations arising from short exact sequences splitting outside S; Ki(oT) is the Grothendieck group of all finitely generated o-torsion free or-modules which are locally projective outside S, with relations arising from short exact sequences. With Cassou-Nogues [3, Corollaire 6.3, p. 231, Queyrut generalized Taylor’s duality result to the wild case by showing that if S is a set of rational primes containing those with a divisor in o wildly ramified in K, then the class of 0 is equal to the class of its dual in K:(U). As in the tame case, this result could be obtained as a consequence of Queyrut’s conjecture concerning the class of 0 in K~(ZI’) [3, p. 81, which was proved in several cases [3]. The proofs of the above results used the description of certain Grothendieck groups as quotients of groups of character functions, as well as results on modules introduced by Swan in [lo].

On Cohomology Groups of Units for Z p -Extensions

Let K/k be a Galois extension of number fields, S a set of finite places on the ground field k, and Es the group of S-units in K. The cohomology groups Hn(K/k, Es), i.e., Hn(Gal(K/k), Es), have been studied for various K/k and S, and found to be an important device to study the arithmetic properties of the extension K/k. In the present paper, we shall make some simple remarks on H'(K/k, Es) in the special case where K is a so-called Zp-extension over a finite algebraic number field k.

Some remarks on the Hasse norm theorem

A substitute for the Hasse norm theorem in Galois extensions of number fields is made more explicit.

Group-theoretic constraints on the structure of the class group

Let K k be a Galois extension of number fields and G its Galois group. By considering the class group of K as a G module we are able to make assertions about its structure once the class number is known. Applications are made to cyclic cubic fields and the 2-class group of cyclotomic fields.

Galois module structure of rings of integers

Let E/F be a Galois extension of number fields with Γ= Gal(E/F) and with property that the divisors of (E:F) are non-ramified in E/Q. We denote the ring of integers of E by 𝒪 E and we study 𝒪 E as a ZΓ-module. In particular we show that the fourth power of the (locally free) class of 𝒪 E is the trivial class. To obtain this result we use Fröhlich’s description of class groups of modules and his representative for the class of ℰ E , together with new determinantal congruences for cyclic group rings and corresponding congruences for Gauss sums.

Normal integral bases and the Galois groups of Galois extensions of number fields.

Normal integral bases and the Galois groups of Galois extensions of number fields.

The Cohomology Groups of Tori in Finite Galois Extensions of Number Fields

Class field theory determines in a well-known way the higher dimensional cohomology groups of the idéies and idèle classes in finite Galois extensions of number fields. At the Amsterdam Congress in 1954 I announced [7] the corresponding result for the multiplicative group of the number field itself, but the proof has never been published. Meanwhile, Nakayama showed that results of this type have much broader implications than had been realized. In particular, his theorem allows us to generalize our result from the multiplicative group to the case of an arbitrary torus which is split by the given Galois extension. We also treat the case of “S-units” of the multiplicative group or torus, for a suitably large set of places S. It is a pleasure for me to publish this paper here, in recognition of Nakayama’s important contributions to our knowledge of the cohomological aspects of class field theory; his work both foreshadowed and generalized the theorem under discussion.