On associative division algebras of prime degree

Abstract

In 1938 Richard Brauer showed2 that, if Z is an associative division algebra of degree five over its center j, there is a field 9 of degree at most twelve over j such that the scalar extension Z X 9 is a cyclic algebra over R. Indeed 9 = Q(aj, a2, (X3), where a, is a root of a quadratic equation over j, a2 is a root of a quadratic equation over (aj), and a3 is a root of a cubic equation over ?(aj, ax2). Since that time there has been no progress in the study of the structure of associative central division algebras of prime degree. In view of Brauer's result it seems reasonable, as a first step in the study of central division algebras Z of prime degree p over j, to consider the case where there is a quadratic field 9 over j such that Za is cyclic. Then Za = Z X has a subfield <3 which is cyclic of degree p over R. The simplest subcase is that where ?3 is actually normal, but, of course, not cyclic over j. We shall treat this case under the assumption that j has characteristic p, and shall prove that then Z is a cyclic algebra over . Our proof proceeds as follows. We are assuming that Z is a central division algebra of prime degree p over j and that 9 = j(t) is a quadratic extension of j. Let J be the automorphism of 9 such that ti = t. Then we may extend J to an automorphism J of Z X 9 such that dJ=d for every d in Z. We are also assuming that there is an element z in Z X 9 such that the field 3 = R (z) is not only cyclic over