Determination of division algebras with 243 elements

Finite nonassociative division algebras (i.e., finite semifields) with 243 elements are completely classified. Nine Knuth orbits were found, two of which are new. All are primitive, and all but the twisted field planes are fractional dimensional.     

Algebraic elements in division algebras over function fields of curves

LetD be a division algebra with centerK a function field of a curveC overk;k(C)=K. We study the maximalk-algebraic subfields ofD. In Theorem 3.1 it is shown that ifD is unramified andC is an elliptic curve thenD contains ak-algebraic splitting field. This enables us to give a new class of counter examples to the Hasse principle for division algebras. The first author is supported by an N.F.W.O. grant. The second author is grateful to the Universities of Antwerp U.I.A. and R.U.C.A. for making it possible for him to do this research.     

Graded division algebras

Mathematische Zeitschrift -     

Cyclic division algebras

A general example of cyclic division algebra is given, based on a construction of Brauer, yielding examples of division algebras of arbitrary prime exponent without proper central subalgebras, and also noncrossed products of arbitrary exponent. This research was supported in part by the U.S.-Israel Binational Science Foundation. An erratum to this article is available at .     

Decomposition of Mal'cev-Neumann division algebras with involution

Deceased, August 31, 1993     

Indecomposable division algebras with a baer ordering

In this paper we construct Baer ordered indecomposable division algebras of index p ⁿ and exponent p ^m for all primes p and for all n m, with n>m≥1 (n≥3 if p=2).     

Selectivity in division algebras

A commutative order in a central simple algebra over a number field is said to be selective if it embeds in some, but not all, maximal orders in the algebra. We completely characterize selective orders in central division algebras, of dimension 9 or greater, in terms of the characterization of selective orders given by Chinburg and Friedman in the quaternionic case.     

On central division algebras

For any integern such that 8|n or for which there exists an odd primeq such thatq ²|n, there is a central division algebra of dimensionn ² over its center which is not a crossed product. The algebra constructed in this paper is the algebraQ(X ₁,…,X)_m, the algebra generated over the rationalQ bym(≧2) generic matrices. To the memory of A. A. Albert This paper was originally presented in November, 1971 for publication elsewhere in a volume in honor of Prof. A. A. Albert on the occasion of his 65th birthday. The volume was never published due to the death of Prof. Albert in June 1972.     

Semidirect product division algebras

SupposeD is a division algebra of degreep over its centerF, which contains a primitivep-root of 1. Also supposeD has a maximal separable subfield overF whose Galois group is the semidirect product of the cyclic groupsC _p C _q , whereq=2, 3, 4, or 6 and is relatively prime top (In particular this is the case whenp is prime ≤7 andD has a maximal separable subfield whose Galois group is solvable.) ThenD is cyclic. The proof involves developing a theory of a wider class of algebras, which we call accessible, and proving that they are cyclic.     

On finite division algebras

A review of the known facts about division algebras of small dimensions over finite fields is given. The cases of dimensions three and four for the commutative algebras are shown to lead to interesting linear spaces of quadrics. This leads to a geometrical classification of the three-dimensional case and a spread of lines of PG(3, q) constructed from any four-dimensional commutative division algebra.     

The centers of generic division algebras with involution

The main goal of this paper is a study of the centers of the generic central simple algebras with involution. These centers are shown to be invariant fields under finite groups in a way analagous to the center of the generic division algebras. The centers of the generic central simple algebras with involution are also described as generic splitting fields (i.e. function fields of Brauer-Severi varieties) over the centers of generic division algebras. Finally, a generic central simple algebra is described for the class of central simple algebras with subfields of a certain dimension. The first author would like to thank the Department of Mathematics of The University of Texas at Austin for its hospitality and the NSF for its support under grant DMS 585-05767. The second author would like to thank the NSF for its support under grants DMS 8303356 and DMS 8601279.     

Transcendence degree of division algebras

We define a transcendence degree for division algebras by modifying the lower transcendence degree construction of Zhang. We show that this invariant has many of the desirable properties one would expect a noncommutative analogue of the ordinary transcendence degree for fields to have. Using this invariant, we prove the following conjecture of Small. Let k be a field, let A be a finitely generated k-algebra that is an Ore domain, and let D denote the quotient division algebra of A. If A does not satisfy a polynomial identity, then GKdim(K) ≤ GKdim(A) − 1 for every commutative subalgebra K of D.