On equivalent number fields with special Galois groups (Q1310154)

The word equivalent in the title means Sonn-equivalence introduced by \textit{J. Sonn} [Isr. J. Math. 52, 239-244 (1985; Zbl 0592.12006)], where it was shown that two equivalent number fields have equal normal closures. It is proved that equivalent fields of degree \(\leq 5\) over \(\mathbb{Q}\) are conjugate. An analogous result is shown to be true if the Galois group of the common normal closure is the full symmetric group. Furthermore, relations between Sonn-, Kronecker- and arithmetical equivalence are discussed. \{Reviewer's remark: A wealth of further material can be found in the author's dissertation (Köln 1992; Zbl 0776.11064)\}.