On generic polynomials

Let K be any field and G be a finite group. Let G act on the rational function field K(xg : g ∈ G )b yK-automorphisms defined by g · xh = xgh for any g, h ∈ G.D enote byK(G) the fixed field K(xg : g ∈ G) G. Noether's problem asks whether K(G) is rational (= purely transcendental) over K. We shall prove that K(G )i s rational overK if G is the dihedral group (resp. quasi-dihedral group, modular group) of order 16. Our result will imply the existence of the generic Galois extension and the existence of the generic polynomial of the corresponding group.

https://msp.org/pjm/2005/222-2/pjm-v222-n2-p05-s.pdf

Noether's problem for dihedral 2-groups

http://www-m11.ma.tum.de/fileadmin/w00bnb/www/people/kemper/kem.generic_few.pdf

Generic Polynomials with Few Parameters

In this paper we study the arithmetic of the Rikuna’s generic polynomial for the cyclic group of order n and obtain a generalized Kummer theory. It is useful under the condition that and ω∈k where ζ is a primitive n-th root of unity and ω=ζ+ζ−1. In particular, this result with ζ∈k implies the classical Kummer theory.

Arithmetic of Rikuna's generic cyclic polynomial and generalization of Kummer theory

We study polynomials giving cyclic extensions over rational function fields with one variable satisfying some conditions. By using them, we construct families of cyclic polynomials over some algebraic number fields. And these families give non-Kummer (or non-Artin-Schreier) cyclic extensions. In this paper, we see that our polynomials have two nice arithmetic properties. One is simplicity: our polynomials and their discriminants have more simple expressions than previous results, e.g. Dentzer (1995), Malle and Mazat (1999) and Smith (1991), etc. The other is a systematic property: if one of our polynomials f gives an extension L/K, then for every intermediate field M we can easily find polynomials giving M/K from f systematically.

/pdf/on-simple-families-of-cyclic-polynomials-3g0uhq49hw.pdf

On simple families of cyclic polynomials

For any cyclic group G of odd order and any field F whose characteristic does not divide the order of G it is possible to find a polynomial which parametrizes all extensions of F with group G. In this paper such a polynomial is explicitly constructed for each such group G which is valid for all such fields F.

Ward Smith Gene

Papers

Generic cyclic polynomials of odd degree