Showing papers on "Abelian extension published in 1978"

Central simple algebras

Abstract: Wedderburn’s factorization of polynomials over division rings is refined and used to prove that every central division algebra of degree 8, with involution, has a maximal subfield which is a Galois extension of the center (with Galois group Z2⊕Z2⊕Z2). The same proof, for an arbitrary central division algebra of degree 4, gives an explicit construction of a maximal subfield which is a Galois extension of the center, with Galois group Z2⊕Z2. Use is made of the generic division algebras, with and without involution.

Galois groups and complex multiplication

Abstract: The Schur problem for rational functions is linked to the theory of complex multiplication and thereby solved. These considerations are viewed as a special case of a general problem, prosaically labeled the extension of constants problem. The relation between this paper and a letter of J. Herbrand to E. Noether (published posthumously) is speculatively summarized in a conjecture that may be regarded as an arithmetic version of Riemann's existence theorem. 0. Introduct.ion. Let F be a perfect field, and F a fixed algebraic closure of F. We consider the finite Galois extensions of F that come from certain types of geometric situations. Let W (vw)V be a cover (finite, flat morphism) of quasi-projective varieties such that W, V, and p(V, W) are defined over F, and W and V are absolutely irreducible. For X, a variety defined over F, we let F(X) be the field of rational functions on X defined over F. Therefore, we have a field extension F(W) over F(V) (by abuse, F(W)/F(V)). If ( V, W) is a separable morphism, then F(W)/F(V) is a finite separable extension, and we obtain (0.1) I G (.''_W) /F' ( V)) -->G (F( IF( V)) 's>tG (F/lF) 1> where: 9W is the smallest Galois extension of F(V) containing F(W) (the Galois closure of F(W)/F(V)); F = F( W, F) is the algebraic closure of F in F(W , and; rest denotes the restriction of elements of the Galois group G( I)/F(V)) to F. We call F the extension of constants obtained from W/V. The problem of the description of G (F/F) arises in several well-known problems. For example. let G be a finite group which we desire to realize as the Galois group of some Galois extension of the rational field Q. Suppose tha: = Q; V is a Zariski open subset of P' (projective n-space); and G(Q(W)/Q( V)), the (geometric) monodromy group is equal to G (a fact that may have come to us from analytic considerations, say in the manner of [Fr, 1]). In this case the limitation theorems of [Fr, 1, ?2] sometimes serve to show Received by the editors February 23, 1976. AMS (MOS) subject classifications (1970). Primary lOB15, lOD25, 12A55, 14D20, 14G05, 14H05, 14H10; Secondary 10M05, 14D05, 14H25, 14H30, 14K22. (1) The research for this paper was partially supported by an Alfred P. Sloan Foundation Grant and by a grant from the Institute for Advanced Study (Princeton) in Spring 1973. r American Mathematical Society 1978

On the central class field ${\rm mod\,}{\germ m}$ of Galois extensions of an algebraic number field

Abstract: Let k be the rational number field, K/k be an Abelian extension defined mod whose degree is some power of a prime l , and let be the module of K belonging to in the sense of Frohlich [1, p. 239].

On the maximal Abelian $l$-extension of a finite algebraic number field with given ramification

Abstract: Let k be a finite algebraic number field and let l be a fixed odd prime number. In this paper, we shall prove the equivalence of certain rather strong conditions on the following four things (1) ~ (4), respectively : (1) the class number of the cyclotomic Z l -extension of k, (2) the Galois group of the maximal abelian l-extension of k with given ramification, (3) the number of independent cyclic extensions of k of degree l, which can be extended to finite cyclic extensions of k of any l-power degree, and (4) a certain subgroup Bk (m, S) (cf. § 2) of k ×/k ×) lm for any natural number m (see the main theorem in §3).

Ideals of an Abelian p -extension of an irregular local field as Galois modules

Abstract: Necessary and sufficient conditions are found under which the Abelian ideal of the p-extension of an irregular local field, treated as a module with operators from the Galois group, is decomposable. In the decomposable case, the decomposition of the ideal into indecomposable terms is found.

On Galois theory using pencils of higher derivations

Abstract: Let L D K be fields of characteristic p =# 0. Assume K is the field of constants of a group of pencils of higher derivations on L, and hence L is modular over K and K is separably algebraically closed in L. Every intermediate field F which is separably algebraically closed in L and over which L is modular is the field of constants of a group of pencils of higher derivations if and only if K(LP') has a finite separating transcendence basis over K for some nonnegative integer e. If p :# 2, 3 and K(LP ) does have a finite separating transcendence basis over K, and F is the field of constants of a group of pencils, then the group of L over F is invariant in the group of L over K if and only if F = K(LP') for some nonnegative integer r.

On homogeneous stochastic processes on compact Abelian groups

Abstract: This article discusses the extension to general compact Abelian groups of some results previously established by R. Roy for the case of the circle and the sphere. Estimators of the covariance function and spectral parameters for a homogeneous stochastic process defined on a compact Abelian group are considered and their properties are derived.