Galois Descent and Generic Splitting Fields

Abstract

Let A be a central simple algebra over F split by a finite dimensional Galois extension field E/F with Galois group G. Then E ⊗ F A=End E V where V is a vector space over E of dimensionality the degree m of A/F. If σ∈G, σ determines the automorphism α σ of End E V that is the identity on A and is σ on E. The α σ form a group and it is clear that A=Inv α the set of fixed points of the α σ . Thus A can be obtained by “Galois descent” from the split central simple algebra End E V. Now α σ has the form \(\ell \rightsquigarrow u_{\sigma}\ell u_{\sigma}^{-1}\) where u σ is a σ-semilinear transformation of V and since u σ is determined up to a multiplier in E * we have u σ u τ =k σ,τ u σ τ for k σ,τ ∈E *. Then the k σ,τ constitute a factor set k from G to E *. The u σ can be used to define a transcendental extension field F m (k) of F in the following way. Let E(ξ)=E(ξ 1,…,ξ m ) where the ξ i are indeterminates and identify the E-subspace ΣE ξ i of E(ξ) with V=ΣEx i , (x 1,…,x m ) a base for V/E. Then the u σ , can be regarded as semilinear transformations of ΣE ξ i and u σ has a unique extension to an automorphism η(σ) of E(ξ)/F such that η(σ)∣E=σ. This restricts to an automorphism η(σ)0 of the subfield E(ξ)0 of rational functions that are homogeneous of degree 0 in the sense that they are quotients of homogeneous polynomials in the ξ’s of the same degree. We have η(σ)0 η(τ)0=η(σ τ)0 (but not η(σ)η(τ)=η(σ τ)). Hence we have the subfield F m (k)=Inv η(G)0 which we call a Brauer field of the central simple algebra A. The field F m (k) is a generic splitting field for A in a sense defined in Section 3.8. Such fields were first studied for quaternion algebras by Witt ([34]) and for arbitrary central simple algebras by Amitsur ([55] and [56]). Further results and a simplification of the theory are due to Roquette ([63] and [64]).