David J. Saltman

Lectures on Division Algebras

Introduction A division algebra is a central simple algebra Azumaya algebras at the generic point The Brauer group Form of matrices Torsion question Galois extensions Crossed products and cohomology Corestriction Orders and regular domains Ramification Specialization and lifting Lattice methods Brauer Severi varieties Generic division algebra Bibliography.

Retract rational fields and cyclic Galois extensions

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 Noethers 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.

Azumaya algebras with involution

One of the highpoints of the theory of central simple algebras as developed in the 1920s and 1930s was the results of Albert concerning simple rings with involution. A part of his results was the characterization of those finite dimensional central simple algebras which admit an involution. This characterization was extended and clarified by Scharlau [8] and Tamagawa (unpublished). It is our intention here to generalize this result in the context of Azumaya algebras over commutative rings. We obtain conditions parallel to the classical ones as to when the equivalence class of an Azumaya algebra contains an algebra with involution. In the last section, we improve and clarify our result in three special cases; Azumaya algebras of rank 4, trivial Azumaya algebras, and Azumaya algebras over connected semilocal rings. Contained in the arguments of this paper is a new proof of the classical result.

Dihedral algebras are cyclic

In his book [1], Albert has a proof that every division algebra of degree 3 is cyclic. In this paper we will generalize this result, and derive the theorem below. Our argument is very close to that of Albert, and arose as part of a close examination of his proof. Fix n to be an odd positive integer, and F a field of characteristic prime to n. Denote by Dn the dihedral group of order 2n. We assume the reader is familiar with the basics of the theory of finite dimensional simple algebras as presented, for example, in Alberts book.

Fields of Definition for Division Algebras

Let

Prime top extensions of division algebra

A

Brauer-Hilbertian fields

be a finite-dimensional division algebra containing a base field

Rank Varieties of Matrices

k

Twisted multiplicative field invariants, Noether's problem, and galois extensions

in its center

Invariant fields of linear groups and division algebras

F