On the Field Intersection Problem of Solvable Quintic Generic Polynomials

Abstract

Let k be a field of characteristic = 2. We survey a general method of the field intersection problem of generic polynomials via formal Tschirnhausen transformation. We announce some of our recent results of cubic, quartic and quintic cases the details of which are to appear elsewhere. In this note, we give an explicit answer to the problem in the cases of cubic and dihedral quintic by using multi-resolvent polynomials. § 1. IntroductionLet G be a finite group, k a field of characteristic = 2, M a field containing k with #M = ∞, and k(t) the rational function field over k with n indeterminates t = (t 1 , . . . , t n ). Our main interest in this note is a k-generic polynomial for G (cf.[DeM83], [Kem01], [JLY02]).Definition. A polynomial f t (X) ∈ k(t)[X] is called k-generic for G if it has the following property: the Galois group of f t (X) over k(t) is isomorphic to G and every G-Galois extension L/M over an arbitrary infinite field M ⊃ k can be obtained as L = Spl M f a (X), the splitting field of f a (X) over M , for some a = (a 1 , . . . , a n ) ∈ M n .be a k-generic polynomial for G. Examples of k-generic polynomials for G are known for various pairs of (k, G) (for example, see [Kem94],

On the Simplest Quartic Fields and Related Thue Equations

“…the simplest cubic case [Mor94], [Cha96], [Oga03], [Kom04], [HM09a], [H]). One of the advantages of using multi-resolvent polynomials is the validity for non-abelian groups (see [HM07], [HM09b], [HM09c], [HM]).…”

On the simplest sextic fields and related Thue equations

“…It is interesting to compare the results of [Oga03], [Kom04], [Kid05] with results given in this section. We note that a method via multi-resolvent polynomials is valid also for non-abelian groups (see [HM07], [HM09b], [HM09c], [HM10c]).…”

A Note on the Field Isomorphism Problem of X3+sX+s and Related Cubic Thue Equations

“…In [3] and [4], the authors gave the following theorem for a field K with char K = 3 as a generalization of Theorem 1 (cf. also [2,Theorem 7] and [5, Theorem 5.1]).…”

“…for such a λ ∈ Z as λ 2 is a divisor of m 3 (4m + 27) 5 . In particular, the primitive solution (x, y) ∈ Z 2 to ( * ) can be chosen to satisfy the relation…”

“…Conversely if there exists a primitive solution (x, y) ∈ Z 2 to ( * ) with a λ ∈ Z such that λ 2 is a divisor of m 3 (4m + 27) 5 , then for the rational number n ∈ Q determined by (1) the splitting fields of f n (X) and of f m (X) over Q coincide, except for the cases of n = 0 and of n = −27/4. For a fixed m ∈ Z \ {0}, furthermore, there exist only finitely many n ∈ Z such that the splitting field of f m (X) and of f n (X) over Q coincide.…”