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Construction of cyclic number fields with prime degree and their Frobenius automorphisms (Algebraic Number Theory and Related Topics 2007)

Generic polynomials are solutions of inverse Galois problem. Generic polynomials are used for the constructions of extension fields, e.g., class fields. In this paper we study the Galois actions and the Frobenius automorphisms of specialization fields of a generic cyclic polynomial given by Cohen [1] and Nakano [4]. Let G be a finite group and k a field. We denote by k(\mathrm{t})[X] the polynomial ring in one variable X whose coefficient ring is the rational function field k(\mathrm{t}) over k with parameters \mathrm{t}=(t_{1}, t_{2}, \ldots, t_{r}) . For