On generic cyclic polynomials of odd prime degree. (English) Zbl 0990.12002

Nakano, Shin

Proc. Japan Acad., Ser. A 76, No. 10, 159-162 (2000).

Let \(\ell\) be a prime number and \(k\) be a field whose characteristic is not \(\ell\). Denote by \(C_\ell\) the cyclic group of order \(\ell\). A polynomial \(g(T_1, T_2, \dots , T_m;X) \in k(T_1, T_2, \dots , T_m)[X]\) having \(C_\ell\) as Galois group over \(k(T_1, T_2, \dots , T_m)\) and parametrizing all \(C_\ell\)-extensions of overfields of \(k\) is called a generic polynomial for \(C_\ell\) over \(k\). For instance, if \(k\) contains the \(\ell\)-th roots of unity \(\mu_{\ell}\), then by Kummer theory, \(X^\ell - T\) is a generic polynomial with one parameter T for \(C_\ell\) over \(k\). Here, the author does not assume that \(\mu_{\ell} \subset k\) and provides a generic polynomial with \(d : = [k(\mu_{\ell}) : k]\) parameters for \(C_\ell\) over \(k\). The proof is based on Theorem 5.3.5 of [H. Cohen, Advanced topics in computational number theory. Graduate Texts in Mathematics, 193 (Springer-Verlag, New York) (2000; Zbl 0977.11056)] of which, in turn, the proof relies on Kummer theory.
As pointed out by the author, different approaches concerning cyclic polynomials of any odd degree over the field of rational numbers \(\mathbb{Q}\) can be found in [R. Dentzer, Commun. Algebra 23, 1593-1603 (1995; Zbl 0822.12001)] and in [G. W. Smith, Commun. Algebra 19, No. 12, 3367-3391 (1991; Zbl 0747.12003)].

Reviewer: Abbas Movahhedi (Limoges)

MSC:

12F10	Separable extensions, Galois theory
12F12	Inverse Galois theory

Keywords:

generic polynomial; cyclic extension

Citations:

Zbl 0977.11056; Zbl 0822.12001; Zbl 0747.12003

References:

[1]	Cohen, H.: Advanced Topics in Computational Number Theory. Grad. Texts in Math., vol. 193, Springer, New York (2000). · Zbl 0977.11056
[2]	Dentzer, R.: Polynomials with cyclic Galois group. Comm. Algebra, 23 , 1593-1603 (1995). · Zbl 0822.12001
[3]	Hashimoto, K., and Miyake, K.: Inverse Galois problem for dihedral groups. Number Theory and Its Applications (eds. Kanemitsu, S., and Gyory, K.). Developments in Math. vol. 2, Kluwer Academic Publ., Dordrecht, pp. 165-181 (1999). · Zbl 0965.12004
[4]	Saltman, D. J.: Generic Galois extensions and problems in field theory. Adv. Math., 43 , 250-283 (1982). · Zbl 0484.12004
[5]	Serre, J.-P.: Topics in Galois Theory. Jones and Bartlett Publ., Boston (1992). · Zbl 0746.12001
[6]	Smith, G. W.: Generic cyclic polynomials of odd degree. Comm. Algebra, 19 , 3367-3391 (1991). · Zbl 0747.12003