Showing papers on "Abelian extension published in 1984"
TL;DR: In this paper, the authors strengthen a recent theorem of Ralph Greenberg [1] concerning L-functions of elliptic curves and anticyclotomic towers, and give an algebraic compositum of all anticycyclotomic extensions of an imaginary quadratic field which are unramified outside the set P of rational primes.
Abstract: In this paper we shall strengthen a recent theorem of Ralph Greenberg [1] concerning L-functions of elliptic curves and anticyclotomic towers. Let K be an imaginary quadratic field and M an abelian extension of K, possibly of infinite degree. We say that the extension M/K is anticyclotomic if M is Galois over Q and if the nontrivial element of Gal(K/Q) acts on Gal(M/K) by inversion. Now fix a finite set P of rational primes and consider the compositum (in an algebraic closure of K) of all anticyclotomic extensions of K which are unramified outside P. We denote this compositum by L and refer to it as the maximal anticyclotomic extension of K unramified outside P. The field L may also be described as the union of all ring class fields of K with conductor divisible only by primes in P. Class field theory shows that Gal(L/K) is isomorphic to the product of a finite group and the group
173 citations
TL;DR: In this paper, the notion of retract rational field extension was defined, and a connection with lifting questions was claimed, which was used to complete the results in [23] and [24] about Noether's problem and generic matrices.
Abstract: In [23], this author began a study of so-called lifting and approximation problems for Galois extensions. One primary point was the connection between these problems and Noether’s problem. In [24], a similar sort of study was begun for central simple algebras, with a connection to the center of generic matrices. In [25], the notion of retract rational field extension was defined, and a connection with lifting questions was claimed, which was used to complete the results in [23] and [24] about Noether's problem and generic matrices. In this paper we, first of all, set up a language which can be used to discuss lifting problems for very general “linear structures”. Retract rational extensions are defined, and proofs of their basic properties are supplied, including their connection with lifting. We also determine when the function fields of algebraic tori are retract rational, and use this to further study Noether’s problem and cyclic 2-power Galois extensions. Finally, we use the connection with lifting to show that ifp is a prime, then the center of thep degree generic division algebra is retract rational over the ground field.
125 citations
38 citations
23 citations
TL;DR: In this article, a general formula for the degree of the reflex field is given, and complete lists of p-structures and reflex fields are provided for (K: Cl ) = 2 n, with n = 3, 4, 5 and 7.
Abstract: A CM-field K defines a triple (C, H, p), where C is the Galois group of the Galois closure of K, H is the subgroup of C fixing K, and p E C is induced by complex conjugation. A '4p-structure" identifies CM-fields when their triples are identified under the action of the group of automorphisms of C. A classification of the p-structures is given, and a general formula for the degree of the reflex field is obtained. Complete lists of p-structues and reflex fields are provided for ( K: Cl ) = 2 n, with n = 3, 4, 5 and 7. In addition, simple degenerate Abelian varieties of CM-type are constructed in every composite dimension. The collection of reflex fields is also determined for the dihedral group C = D2n, with n odd and H of order 2, and a relative class number formula is found.
22 citations
TL;DR: In this paper, it was shown that a finitely generated almost solvable group with decidable theory is almost Abelian, and a decidable decidable Abelian group is shown to be Abelian as well.
Abstract: It is shown that a finitely generated almost solvable group with decidable theory is almost Abelian. Bibliography. 13 titles.
22 citations
TL;DR: The main difference between ordinary and modular representation theory of finite groups is the existence of nonprojective modules in the modular representation algebra as discussed by the authors, which is the case of the projective cover of M over the group algebra kG.
16 citations
Journal Article•
TL;DR: In this article, the conditions générales d'utilisation (http://www.compositio.org/conditions) of the agreement with the Foundation Compositio Mathematica are defined.
Abstract: © Foundation Compositio Mathematica, 1984, tous droits réservés. L’accès aux archives de la revue « Compositio Mathematica » (http: //http://www.compositio.nl/) implique l’accord avec les conditions générales d’utilisation (http://www.numdam.org/conditions). Toute utilisation commerciale ou impression systématique est constitutive d’une infraction pénale. Toute copie ou impression de ce fichier doit contenir la présente mention de copyright.
11 citations
TL;DR: In this paper, it was shown that the absolute Galois group of a field of characteristic 0 and t an indeterminate is a semi-direct product of a free profinite group.
Abstract: Let K be a field of characteristic 0 and t an indeterminate. It is shown that the absolute Galois group of K(t) is the semi-direct product of a free profinite group with the absolute Galois group of K.
11 citations
TL;DR: The theory of separable mixed abelian groups has been studied in this paper, where the authors consider the case of groups with separable separable abelians and separable groups with mixed groups.
Abstract: (1984). The theory of separable mixed abelian groups. Communications in Algebra: Vol. 12, No. 15, pp. 1813-1834.
TL;DR: In this article, it was shown that for almost all e-tuples σ = (σ 1, σ, σ e ) of elements of the abstract Galois group G(K) of K and for all elliptic curves E defined over the prime field, the following results hold.
Abstract: The following theorem is proved in [2[. Let K be a finitely generated field over its prime field. Then for almost all e-tuples σ = (σ 1 , …, σ e ) of elements of the abstract Galois group G(K) of K and for all elliptic curves E defined over the following results hold.
TL;DR: In this article, a Galois extension of a local or global field is defined as a finite extension of k with g = Gal (K/k) and g is called central if it lies inside the centre of Gal(L/K).
Abstract: Let k be either a local or a global field, and K be a finite Galois extension of k with g = Gal (K/k). Let L be a Galois extension of K which is also Galois over k. Such an extension is called central if Gal(L/iT) lies inside the centre of Gal(L/K). Clearly L is abelian over K. Next set L* = L∩K · kab where kab is the maximal abelian extension of k in its algebraic closure. This is the genus field of L over K/k.
TL;DR: For fields of algebraic numbers for which the Galois group of a maximal l-extension with ramification at the divisors of l is a Demushkin group, see as discussed by the authors.
Abstract: Examples are given of fields of algebraic numbers for which the Galois group of a maximal l-extension with ramification at the divisors ofl is a Demushkin group.
TL;DR: In this article, the authors define the genus field resp. the central class field of K with respect to M/k, which is the composite of K and the maximal abelian extension over k contained in M. The field is the maximal Galois extension of k contained within M satisfying the condition that the Galois group over K is contained in the center of that over k.
Abstract: Let K be a finite Galois extension of an algebraic number field k with G = Gal ( K/k ), and M be a Galois extension of k containing K . We denote by resp. the genus field resp. the central class field of K with respect to M/k . By definition, the field is the composite of K and the maximal abelian extension over k contained in M . The field is the maximal Galois extension of k contained in M satisfying the condition that the Galois group over K is contained in the center of that over k . Then it is well known that Gal is isomorphic to a factor group of the Schur multiplicator H -3 (G, Z ), and is isomorphic to H -3 (G, Z ) when M is sufficiently large. In this case we call M abundant for K/k (See Heider [3, § 4] and Miyake [6, Theorem 5]).
TL;DR: In this article, the capitulation problem of the transfer-kernel for Galois field towers was studied in certain class-formations of number fields, and the cohomology of the S-idel class group of the maximal extension of k unramified outside a finite set S ⊃ S∞ of primes was analyzed.
Abstract: The capitulation problem concerns the determination of the transfer-kernel Ke((G(L¦k)ab→G(L¦K)ab) for Galois field towers L¦K¦k in certain classformations of number fields. 3 special instances are dealt with: (1) The capitulation kernel can be described in terms of ray knots if L is a ray class field over K. (2) lyanaga's application of the genus conductor is extended to the non-abelian case. (3) The cohomology of the S-idel class group of the maximal extension of k unramified outside a finite set S ⊃ S∞ of primes can be expressed by the cohomology of the S-Leopoldt-kernel. This leads to a description of the respective capitulation kernel and moreover of the Schur multiplier of the Galois group of the maximal p-extension unramified outside p (p≠2) in terms of the usual Leopoldt-kernel.
TL;DR: In this article, a method for constructing examples and pointing out directions for further classification of real fields with simple absolute Galois groups has been presented, which led to the discovery of hereditarily pythagorean fields.
Abstract: In his survey in this volume, Becker mentions how his search for real fields with simple absolute Galois groups led him to the discovery of hereditarily pythagorean fields. This note will illustrate his remark. It will serve two purposes : Provide a method of constructing examples and point out directions for further classification of real fields. Proofs of the results will appear elsewhere. Let k be a real field, Q its algebraic closure and G = Gal(û|fc), its absolute Galois group. G can be very large and so one looks for simple identifiable types of small closed subgroups H of G containing involutions so that the corresponding fixed fields of H are all real.
TL;DR: In this paper, the authors investigated the structure of 0 as an or-module by comparing 0 with its dual, Horn@, 0, and showed that the class of 0 is equal to its dual in K:U.
TL;DR: In this article, the authors dealt with symplectic spaces over noncommutative rings of a special type, in which unique division by 2 is impossible, and obtained a description of the multiplicative group of certain local fields that play an important role in determining the Galois group of the algebraic closure of extensions of the field of 2-adic numbers.
Abstract: This paper deals with symplectic spaces over noncommutative rings of a special type, in which unique division by 2 is impossible. An application of the results obtained is a description of the multiplicative group of certain local fields that play an important role in determining the Galois group of the algebraic closure of extensions of the field of 2-adic numbers.
TL;DR: In this paper, the structure of ideal class groups of number fields is investigated in the following three cases: (i) Abelian extensions of number field whose Galois groups are of type (p, p); (ii) non-Galois extensions Q(p d 0 3,p d 1 3 ) of degree p2 over Q; (iii) dihedral extensions of degree 2n + 1 over Q.