Showing papers on "Abelian extension published in 1990"

Abelian Groups, Fields and Vector Spaces 1

Abstract: The articles [3], [1], and [2] provide the notation and terminology for this paper. We consider group structures which are systems 〈 a carrier, an addition, a reverse-map, a zero 〉 where the carrier is a non-empty set, the addition is a binary operation on the carrier, the reverse-map is a unary operation on the carrier, and the zero is an element of the carrier. In the sequel GS denotes a group structure. Let us consider GS. An element of GS is an element of the carrier of GS. Next we state a proposition (1) For every element x of the carrier of GS holds x is an element of GS. We now define three new functors. Let us consider GS. The functor 0GS yields an element of GS and is defined by: 0GS =the zero of GS. Let x be an element of GS. The functor −x yielding an element of GS, is defined by: −x =(the reverse-map of GS)(x). Let y be an element of GS. The functor x + y yielding an element of GS, is defined by: x + y =(the addition of GS)(x, y). Next we state three propositions: (2) 0GS =the zero of GS. (3) For every element x of GS holds −x =(the reverse-map of GS)(x). (4) For all elements x, y of GS holds x + y =(the addition of GS)(x, y).

Finite simple abelian algebras are strictly simple

Abstract: A finite universal algebra is called strictly simple if it is simple and has no nontrivial subalgebras An algebra is said to be Abelian if for every term t(x,yp) and for all elements a, b, c, d, we have the following implication: t(a, c) = t(a, d) -t(b, c) = t(b, d) It is shown that every finite simple Abelian universal algebra is strictly simple This generalizes a well-known fact about Abelian groups and modules

The K-admissibility of SL(2, 5)

Abstract: Let K be a field and let G be a finite group. G is K-admissible if there exists a Galois extension L of K with G=Gal(L/K) such that L is a maximal subfield of a central K-division algebra. We characterize those number fields K such that H is K-admissible where H is any subgroup of SL(2, 5) which contains a S2-group. The method also yields refinements and alternate proofs of some known results including the fact that A5 is K-admissible for every number field K.

Generalized Class Field Theory

Abstract: This paper is a survey of the X-theoretic generalization of class field theory. For a field K, let K be a maximal abelian extension of K, that is, the union of all finite abelian extensions of K in a fixed algebraic closure of K. The classical local (resp. global) class field theory says that if K is a finite extension of the p-adic (resp. rational) number field Qp (resp. Q), the Galois group Gal(K /K) is approximated by the multiplicative group K (resp. the idele class group CK), and via this approximation, we can obtain knowledge on abelian extensions of K. In §1 (resp. §2), we give a £>theoretic generalization of the classical local (resp. global) class field theory. There finite extensions of Qp (resp. Q) are replaced by "higher dimensional local fields" (resp. finitely generated fields over prime fields), and the group K (resp. CK) is replaced by Milnor's JC-group K^(K) (resp. by the JC^-idele class group), where n is the "dimension" of K. In §3, we discuss some other aspects of generalizations of local class field theory. In §4, we discuss generalizations of the classical ramification theory to higher dimensional schemes.

The Fundamental Theorem of Galois Theory

Abstract: Given a Galois extension E / F, the fundamental theorem will show a strong connection between the subgroups of Ga1(E / F) and the intermediate fields between F and E.

The Galois group of a radical extension of the rationals

Abstract: The Galois group of the splitting field of an irreducible binomialx2e−a overQ is computed explicitly as a full subgroup of the holomorph of the cyclic group of order 2e. The general casexn−a is also effectively computed.

A note on the quaternion group as Galois group

Abstract: The occurrence of the quaternion group as a Galois group over certain fields is investigated. A theorem of Witt on quaternionic Galois extensions plays a key role. In [9, ?6] Witt proved a theorem characterizing quaternionic Galois extensions. Namely, he showed that if F is a field of characteristic not 2 then an extension L = F(.la , v's), a, b E F, of degree 4 over F can be embedded in a Galois extension K of F with Gal(K/F) H8 (the quaternion group 2 2 2 . of order 8) if and only if the quadratic form ax + by + abz is isomorphic 2 2 2 to x + y + Z . In addition he showed how to explicitly construct the Galois extension from the isometry. An immediate and interesting consequence of this is the fact that H8 cannot be a Galois group over any Pythagorean field. In this note Witt's theorem is used to obtain additional results about the existence of H8 as a Galois group over certain fields. If F is a field (of characteristic not 2) with at most one (total) ordering such that H8 does not occur as a Galois group over F then the structure of the pro-2-Galois groups GF(2) = Gal(F(2)/F), G = Gal(F I/F) (where F(2) and F are the F py py ~~~~~~~~~~~~~~~~~py quadratic and pythagorean closures of F) are completely determined. Moreover it is shown that for any field F of characteristic not two, H8 occurs as a Galois group over F iff H8 is a homomorphic image of Gpy iff the dihedral group D8 of order 8 is a homomorphic image of Gpy . In what follows all fields have characteristic different from 2. If a an E F = F \ {O} then q = (a1, ... , an) denotes the quadratic form with orthogonal basis el, ... ,en and q(ei) = ai . The value set of q is D(q) = {a E FPq(x) = a for some x}. Any unexplained notations and terminology about quadratic forms can be found in [4]. Lemma 1. For a field F with -1 F2 and Ip,p2l > 2 the following are equivalent: (1) The level (stufe), s(F), of F is two. Received by the editors March 24, 1989. 1980 Mathematics Subject Classification (1985 Revision). Primary 11 E8 1, 1 2F1 0; Secondary, 12D15. Supported in part by NSA research grant No. MDA 904-88-H-2018. ( 1990 American Mathematical Society 0002-9939/90 $1.00 + $.25 per page

A note on relations between the zeta-functions of Galois coverings of curves over finite fields

Abstract: Abstract Let C be a complete irreducible nonsingular algebraic curve defined over a finite field k. Let G be a finite subgroup of the group of automorphisms Aut(C) of C. We prove that certain idempotent relations in the rational group ring Q[G] imply other relations between the zeta-functions of the quotient curves C/H, where H is a subgroup of G. In particular we generalize some results of Kani in the special case of curves over finite fields.

On the non-existence of abelian conditions governing solvability of the - 1 Pell equation.

Abstract: It has been known since Lagrange that the + 1 equation has infmitely many Solutions, which can all be found using the arithmetic associated with the continued fraction of ]fd. (See [10]). Using the same continued fraction one can also determine whether or not the — l equation is solvable; indeed, the — l equation has a solution iff the length of a primitive period of the continued fraction of J/öf is odd.

On Some Properties of Free Abelian Extensions

Abstract: Let be an arbitrary group. Present it as a factor group of a free group , . The extension is said to be a free abelian extension of the group (it is free in the category of extensions of by all possible abelian groups). The author continues his study of the integral homology groups . The main result is that for any the exponent of the torsion subgroup of the group divides (as usual, is the ring of 2-rational numbers). At the end of the paper the author formulates a number of conjectures on the homology of groups of the form . The notion of homological identity of a group is introduced, and the problem of describing the homological identities of free solvable and free nilpotent groups is posed.Bibliography: 9 titles.

Canonical structure of theO(3) nonlinear sigma model with a topological term

Abstract: We canonically quantize theO(3) nonlinear sigma model in 1+1 spacetime with a topological term. TheO(3) current algebra is obtained and found to have an abelian extension similar to the 2+1 dimensional model with a Hopf term. The current algebra is independent of the parameter μ, which is the coefficient of the topological term. At μ=π we show the equivalence of this algebra withk→∞ limit of the algebra ofSU(2)⊗SU(2) invariant WZW model. Following the prescription of nonlinear quantum mechanics, we resolve the operator ordering ambiguities and obtain the quantum Hamiltonian.

The Lutz filtration as a Galois module in an extension without higher ramification

Abstract: One considers the structure of the group of the points of a formal group and its Lutz filtration as a Galois module in an extension without higher ramification of a local field. Making use, on one hand, of Honda's theory on the classification of formal groups over complete local rings and, on the other hand, of a generalization to formal groups of the Artin-Hasse function, one constructs effectively an isomorphism between the group of points and some given additive free Galois module. In particular, in the multiplicative case one gives a new effective proof of Krasner's theorem on the normal basis of the group of principal units of a local field in extensions without higher ramification.

The Galois Group

Abstract: We now set up an analogy with symmetries of polygons in the plane even though some of the algebraic analogues have not yet been defined.

Locally supersolvable extensions of Abelian groups

Abstract: An extension E of an Abelian group A by a locally supersolvable group is studied under the assumption that A has a finite series of subgroups normal in E, each of whose factors satisfies one of the following conditions: 1) min-E; 2) max-E; 3) the factor has finite rank and is torsion-free. It is proved that if E induces in the factors of the series in A hypercyclic groups of automorphisms and A is the locally supersolvable coradical of E, then A is complemented in E and all complements are conjugate in E. An analog of this result for locally nilpotent extensions of Abelian groups is also noted.

Generating functions for the numbers of abelian extensions of a local field

Abstract: The aim of this paper is to give an explicit formula for the numbers of abelian extensions of a p-adic number field and to study the generating function of these numbers. More precisely, we give the number of abelian extensions with given degree and ramification index, and the number of abelian extensions with given degree of any local field of characteristic zero. Moreover, we give a concrete expression of a generating function for these last numbers.