Noncyclic division algebras

Abstract

An associative algebra whose center is a field will be called centrally finite if it is finite dimensional over its center. An algebraic extension of commutative fields F c K has been called Galois provided that K is a finite, normal, separable extension of F. An extension F c K of fields is cyclic provided it is a Galois extension whose Galois group G(K/F) is cyclic. If D is a division ring that is centrally finite over F and K c D any maximal subfield, then the F-vector space dimensions satisfy [D:F]=[,K:F]2=[G(K/F)] 2. A simple algebra A that is centrally finite over its center F will be called cyclic if (i) there exists a maximal subfield K ~ A such that F ~ K is cyclic, and (ii) [,A :F] = [K :F] 2. In the construction of centrally finite noncrossed product division rings in [-6] and [14], a certain class of noncyclic crossed product division algebras had to be constructed ([-6; p. 412, Theorem 3], or [14; p. 102, Theorem 1]). The centers of these latter division rings had to be formal Laurent series fields in a finite number of variables over an algebraically closed field. The purpose of this note is to construct a new class of noncyclic crossed products. Their centers are more general than in the construction in [6]. The method of proving noncyclicity is not new, it is Albert's. In actually proving that the algebras are noncyclic, several new classes of division rings will be constructed, which may be of independent interest. The main difficulty is not in defining these algebras, but in actually proving that they are division rings. The point of this note is that by using an algebra D that is different than Albert's original one, a straightforward, simple proof which is different from Albert's original proof ([-1, 2], and [12, 9, 13]) that D is a noncyclic division ring can be given. Also, in [2] the center F was a formally real field ([2; p. 449, bottom]), while the present proof holds more generally for centers F such that ] / I r In particular, the construction may be used in case char F + 0.