Abstract
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.
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References
Draxl, P.K. [83]: Skew Fields. London Math. Soc. Lecture Notes 83, Cambridge University Press 1983
Ore, O. [33]: Theory on noncommutative polynomials. Ann. Math. 34 (1933), 480–508
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© 1996 Springer-Verlag Berlin Heidelberg
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Jacobson, N. (1996). Skew Polynomials and Division Algebras. In: Finite-Dimensional Division Algebras over Fields. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-02429-0_1
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DOI: https://doi.org/10.1007/978-3-642-02429-0_1
Publisher Name: Springer, Berlin, Heidelberg
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