Covering subgroups of groups and Kronecker classes of fields (Q1117268)

Two finite extensions K and \(K'\) of an algebraic number field k are called Kronecker-equivalent over k if the sets of all prime ideals in k having a prime divisor of relative degree 1 in K resp. \(K'\) have finite symmetric difference. \textit{W. Jehne} [J. Number Theory 9, 279-320 (1977; Zbl 0392.12003)] has shown that the equivalence classes are infinite in most cases. \textit{J. Saxl} [J. Lond. Math. Soc., II. Ser. 38, No.2, 243- 249 (1988; Zbl 0663.12010)] has shown that the Kronecker class consists of one element in the case when \([K:k]=2\). In the present paper the following interesting result is demonstrated: Theorem 4.1. Let \(K'\) be a Galois extension of the algebraic number field of k of degree 4, and assume that K is an atomic extension of \(K'\) such that K and \(K'\) are Kronecker equivalent over k. Then \([K:K']=3\) and the extension is not Galois. A result of W. Jehne (loc. cit.) immediately reduces the proof to a problem on (Galois-)groups covered by conjugates of a proper subgroup. A purely group-theoretical argument is then used to determine a list of all possible Galois groups (theorem 4.3).