Factorization over finitely generated fields

This paper considers the problem of factoring polynomials over a variety of domains. We first describe the current methods of factoring polynomials over the integers, and extend them to the integers mod p. We then consider the problem of factoring over algebraic domains. Having produced several negative results, showing that, if the domain is not properly specified, then the problem is insoluble, we then show that, for a properly specified finitely generated extension of the rationals or the integers mod p, the problem is soluble. We conclude by discussing the problems of factoring over algebraic closures.