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Let k be a field of characteristic = 2. We give an answer to the field intersection problem of quartic generic polynomials over k via formal Tschirnhausen transformation and multi-resolvent polynomials.

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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],