Finite-Dimensional Division Algebras over Fields

We assume the reader is familiar with the standard ways of constructing “simple” field extensions of a given field F, using polynomials. These are of two kinds: the simple transcendental extension F(t), which is the field of fractions of the polynomial ring F[t] in an indeterminate t, and the simple algebraic extension F[t]/(f(t)) where f(t) is an irreducible polynomial in F[t]. In this chapter we shall consider some analogous constructions of division rings based on certain rings of polynomials D[t; σ, δ] that were first introduced by Oystein Ore [33] and simultaneously by Wedderburn. Here D is a given division ring, σ is an automorphism of D, δ is a σ-derivation (1.1.1) and t is an indeterminate satisfying the basic commutation rule for a∈D. The elements of D[t; σ, δ] are (left) polynomials where multiplication can be deduced from the associative and distributive laws and (1.0.1) (cf. Draxl [83]). We shall consider two types of rings obtained from D[t; σ, δ]: homomorphic images and certain localizations (rings of quotients) by central elements. The special case in which δ=0 leads to cyclic and generalized cyclic algebras. The special case in which σ=1 and the characteristic is p≠0 gives differential extensions analogous to cyclic algebras.

In this chapter we consider the main classical results of the structure theory of central division algebras and more generally of central simple algebras over arbitrary fields. These center on two closely related problems: the determination of the algebras and the structure of the Brauer group Br(F) of a field F.

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 _στ 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]).

A p-algebra is a central simple algebra A over a field of characteristic p>0 such that [A]∈Br_p(F), the p-th component of Br(F), that is, the exponent of A is a power of p (equivalently, the index is a power of p). The structure theory of these algebras was developed by Albert in the thirties and was presented in an improved form in Chapter VII of his Structure of Algebras (see also Teichmüller [36]). The culminating result of Albert’s theory is that, in his terminology, any p-algebra is cyclically representable, that is, is similar to a cyclic algebra. This is proved in two stages: the first, in which it is shown that any p-algebra is similar to a tensor product of cyclic algebras and the second, in which it is proved that the tensor product of two cyclic p-algebras is cyclic. The first of these results is proved in Section 4.2. Purely inseparable splitting fields play an important role in the theory. We recall that K/F is purely inseparable if K/F is algebraic and its subfield of separable elements coincides with F, or equivalently, for any a∈K there exists an e≥0 such that \(a^{p^{e}}\in F\). The exponent of K/F is e<∞ if there exists an e such that \(a^{p^{e}}\in F\) for all a∈K and e is minimal for this property. If [K : F]=n<∞ then K has exponent e such that p ^e≤n. (See Section 8.7 of BA II for these results.)

The study of algebras with involution and especially the simple ones first arose in Albert’s work on the multiplication algebras of Riemann matrices. This led him to the systematic study of algebras with involution for their own sake. His results were published in Structure of Algebras ([39]). A part of this chapter is devoted to presenting these results and recent extensions of them.