Skew Polynomials and Division Algebras

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 1.0.1 $$ta=(\sigma a)t+\delta a$$ for a∈D. The elements of D[t; σ, δ] are (left) polynomials 1.0.2 $$a_0+a_1t+\cdots +a_nt^n,\qquad a_i\in D$$ 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.