On generic polynomials (Q1870087)

In this paper the authors examine over arbitrary fields the possible implications among the concepts due to D. Saltman of generic Galois extensions, retract rational extensions, the lifting property for Galois extensions, the notions due to G. Smith of generic polynomials, and of descent generic polynomials due to F. DeMeyer. As a consequence of work by Ledet and Kemper, it is known that for infinite fields, the existence of a generic Galois extension, the lifting property, retract rationality of \(F(x_\sigma|\sigma\in G)^G\), the existence of a generic polynomial, and the existence of a descent generic polynomial are all equivalent. The purpose of this paper is to study the situation over (possibly) finite fields. The main results of this paper show that the theory breaks into two parts when the base field is not assumed to be infinite. In the first part, all extensions which appear in the theory are restricted to be fields, and subgroups of the Galois group are not included (Theorem 1). In the second part extensions are permitted to include direct sums of fields, and subgroups of the Galois group are included (Theorem 2). The authors were unable to decide whether the existence of a generic polynomial for Galois field extensions with a given group \(G\) and a Galois extension with group \(G\) generic for Galois ring extensions are equivalent (Theorem 7). The following results are abbreviated forms of the authors' main results. Theorem 1. The following statements are equivalent. (1) There is a generic polynomial for Galois extensions \(N/K\) of \(F\) with group \(G\). (2) The pair \((G,F)\) satisfies the lifting property for Galois field extensions. (3) \(F(x_\sigma |\sigma \in G)^G\) is retract rational. (4) There is a generic Galois extension for field extensions \(N/K\) of \(F\). Theorem 2. The following statements are equivalent. (1) There is a generic polynomial for Galois ring extensions \(L/K\) with group \(G\) for any field extension \(K\) of \(F\). (2) There is a descent generic polynomial for \(G\) over \(F\). Theorem 7. Let \(F\) be a field and \(G\) a finite group. Consider the following statements. (1) There is a descent generic polynomial for \(G\) over \(F\). (2) There is a generic Galois extension for \(G\) over \(F\). (3) There is a generic polynomial for \(G\) over \(F\). Then (1) implies (2) implies (3), but (2) does not imply (1) and (3) does not imply (1).