On stem extensions of \(S_ n\) as Galois group over number fields
Many years ago Emmy Noether asked whether every finite group can be realized as a Galois group over the rational number field. The present paper deals with the problem of the realizability of the finite stem extensions of the symmetric group \(S_ n\) as a Galois group over number fields. A stem extension of a finite group G is a central extension \(1\to A\to E\to G\to 1\), in which \(A\subset E'\) (the commutator group of E). The author gives a wide class of polynomials with such Galois groups. For example, if \(n\equiv 0,1,2,\quad or\quad 3\quad (mod 8),\) then every finite stem extension of \(S_ n\) occurs as a Galois group over every number field K. Moreover, if \(i\in K\), then every finite stem extension of \(S_ n\) can be realized as a Galois group over K for every value of n.
scientific article; zbMATH DE number 4081686
scientific article