The basic definitions and Baer's classificationWe repeat some definitions and basic results of [9] for the convenience of the reader. The Kronecker set D(K\k) of a finite extension K\k of a number field k consists of all prime divisors of k having a prime divisor in K of first relative degree over k. Two extensions K, K' of k are said to be Kronecker equivalent K~kK' if D{K\k) = D{K'\k) (equality up to finite sets); St = Rjc(K) denotes the Kronecker class of K. Kronecker equivalent fields can be considered to share the same 'weak decomposition behaviour'.As to the invariants one can show that all minimal fields L in a Kronecker class St have the same galois hull M over k\ M is called the galois hull M(St) of St, and its galois group over k is called the galois group G(St) of St. Further invariants are the finite graph 9JI (with respect to inclusion) of all fields of ft contained in M(St), called the socle graph of R, and the set £(ft) of all minimal fields in St. Of interest are only the nonconjugate fields in those sets under the action of G(St); the number of nonconjugate fields in the socle graph is called the socle number (Ji k {St) of St, and the number of non-conjugate minimal fields in St is called the width co k (St) of St. Obviously a>(St) < fx(St).We are mainly concerned with these two invariants, or closely related ones, in this paper. In [9] it has been shown that many of the possible cases actually occur for suitable number fields, for instance: both invariants are 1; both are different and different from 1; or, one of the main results, a)(St) = fx(St) can be an arbitrary high prime power.Kronecker equivalence can easily be translated into a group-theoretical equivalence relation by means of the Artin symbol. Let M \ k be a galois extension containing the given extensions K, K' of k, let G be the galois group of M\k, and let U, U' be the fixed groups in G of K, K' respectively.
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