Thomas McKenzie

On generic polynomials

Abstract We examine over arbitrary fields the possible implications among the concepts due to D. Saltman of generic Galois extension, retract rational extension, the lifting property for Galois extensions, the notions due to G. Smith of generic polynomial, and of descent generic polynomial due to F. DeMeyer.

Toward an arithmetic of polynomials

SummaryForR a commutative ring, which may have divisors of zero but which has no idempotents other than zero and one, we consider the problem of unique factorization of a polynomial with coefficients inR. We prove that, if the polynomial is separable, then such a unique factorization exists. We also define a Legendre symbol for a separable polynomial and a prime of commutative ring with exactly two idempotents in such a way that the symbols of classical number theory are subsumed. We calculate this symbol forR = Q in two cases where it has classically been of interest, namely quadratic extensions and cyclotomic extensions. We then calculate it in a situation which is new, namely the so called generalized cyclotomic extensions from a paper by S. Beale and D. K. Harrison. We study the Galois theory in the general ring situation and in particular define a category of separable polynomials (this is an extension of a paper by D. K. Harrison and M. Vitulli) and a cohomology theory of separable polynomials.