Abelian Galois Group Research Articles

Kummer extensions of rings

The theory of Kummer extensions of commutative rings is constructed, generalizing the theory of Kummer extension fields. A Galois extension S/R with Abelian Galois group of degreen is called Kummerian if in the group of invertible elements of the ringR there is a cyclic subgroup of ordern such that for any , θ≠1 element 1-θ is invertible inR. Any cyclic Kummerian extension can be obtained by means of a suitable factorization of the tensor algebra over a finitely generated protectiveR -module of rank 1 (an arbitrary Kummerian extension is a tensor product of cyclic ones). The group of equivalence classes of Kummerian extensions with fixed Galois group is studied.

Commutative Abelian Galois extensions of a Banach algebra

Let A be a commutative unital Banach algebra with connected maximal ideal space X. We show that the Gelfand transform induces an isomorphism between the group of commutative Galois extensions of A with given finite Abelian Galois group, and the corresponding group of extensions of C( X). This result is applied, when X is sufficiently nice, to construct a separable projective finitely generated faithful Banach A-algebra whose maximal ideal space is a given finitely fibered covering space of X.

Abelian extensions in picard - Vessiot theory

A classification is given of Pcard-Vessiot extensions with an Abelian Galois group for a partial differential field of characteristic zero with an algebraically closed field of constants.

Remark on a characterization of certain ring class fields by their absolute Galois group

where A(co) stands for the discriminant of the complex lattice generated by 1 and to and g2(u) is the Weierstrass invariant of this lattice. If ti=Q(\/ — D) is an imaginary quadratic number field, it is well known from the theory of class fields with complex multiplication that the singular values j(a), where a£u, la >0, generate algebraic number fields which are abelian over fi, namely the so-called ring class fields. Detailed references for the literature on the theory of ring class fields may be found in the report of Deuring [l]. In the rather extensive theory of ring class fields it is shown that ®Uia))/Q iS a normal extension with its Galois group © being an extension of the abelian Galois group s of Q(j(a))/Cl, completely determined by the relations

Abelian Galois groups

The question of the existence of noninner, nonouter Abelian Galois groups of noncommutative rings seems not to have been considered previously. Amitsur [1 ] may have come closest when he constructed noninner, nonouter cyclic division ring extensions. Although these extensions are finite dimensional division ring extensions of Galois subrings corresponding to cyclic groups of automorphisms, nevertheless, as will be shown below, their Galois groups need not be Abelian. The question is made more critical by results obtained below which imply for any finite dimensional division algebra, and, furthermore, for any full matrix ring over it having center 7 GF(4), that any Abelian Galois group is inner or outer. Curiously enough Galois groups of the required kind do exist for some matrix rings over GF(4). Affirmative results are obtained also for certain infinite dimensional division algebras first constructed by Kothe [3 ], and their associated matrix rings. For these rings a general procedure for obtaining noninner, nonouter Abelian Galois groups is discussed.