An Example in Central Division Algebras

Abstract

Given r, n two relatively prime integers, and m be the minimal integer such that r ^m ≡ 1 (mod n). We construct, for each s = r ^t, (t, m) = 1 a central division algebra A _S of dimension m ² over a field of rational function K(X) in a commutative indeterminate X, and K a finite extension of the rationals Q. These algebras have peculiar properties: The ring of rational functions A _s (ξ) over a commutative indeterminate ξ-are isomorphic as algebras, but belong to different Brauer classes over their center K(X, ξ) (theorem 4.1). (b) The division algebras A _s are non-isomorphic (as rings) except that A _s ≅A _v if sv ≡ l(mod m), yet for m≠ 2-they belong to different Brauer classes, (c) Nevertheless, there exist a chain of ring isomorphism maps into: (but not onto!) A _r→A ^v1 _r →…→A ^Vl _r →A _r where {v _i } is a set of integers relatively prime to m. (Corollary 4.4). (d) By Tsen’s theorem, the algebras A _S have splitting fields of the form L(X) where L/K is an algebraic extension. (Theorem 5.1) For all these algebras there exists an algebraic extension F⊃K such that F(X) is a common maximal subfield of all A _S , and L(X) splits A _s if and only if L⊇F. Further relations between these divisions algebras are discussed (§5).