On simple families of cyclic polynomials

Abstract

Abstract. 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.

HT90 and “simplest” number fields

“…Taking x = cot(θ), we see that the zeroes of P n (a, x) are permuted cyclically by the linear fractional transformation λ in (3.6). The results in this section are thus very similar to those in [31], but are less general. We assume our parameter to be in the complex field C. This allows us to "cheat" by invoking properties of cot(θ) as a periodic meromorphic function, and to exploit the fact that λ corresponds to adding a division point to the argument of this function.…”

On the Simplest Quartic Fields and Related Thue Equations

“…For n ≥ 3, Rikuna [Rik02] constructed one-parameter families of cyclic polynomials of degree n over K with char K | n and K ∋ ζ + ζ −1 where ζ is a primitive n-th root of unity, and f s (X) may be obtained the quartic case n = 4 of Rikuna's cyclic polynomials (see also [Miy99], [HM99]). An answer to the field isomorphism problem to Rikuna's cyclic polynomials was given by Komatsu [Kom04] as a generalization of Kummer theory (cf.…”

Arithmetic Properties of Generalized Rikuna Polynomials

“…In [11], Rikuna introduced a one-parameter family of polynomials with wide-ranging applications to arithmetic. In particular, let > 2 be a fixed positive integer (not necessarily prime) and K a field of characteristic coprime to that does not contain a primitive -th root of unity.…”

“…In particular, let > 2 be a fixed positive integer (not necessarily prime) and K a field of characteristic coprime to that does not contain a primitive -th root of unity. Fix an algebraic closure K of K and fix a primitive -th root of unity ζ ∈ K. Assume further that ζ + := ζ + ζ −1 ∈ K. Following [11], define the polynomials p(x), q(…”

“…Rikuna originally introduced r(x, t) as a method for creating cyclic Galois extensions of a base field that are not given by Kummer or Artin-Schreier theory (hence the requirement that ζ ∈ K). He proved in [11] that for all ≥ 3, r(x, t) is irreducible over K(t) and has Galois group Z/ over K(t). It was then shown in [6] that when is odd r(x, t) is generic in the sense that every Z/ -extension of L with L ⊃ K is obtained as a specialization of r(x, t).…”