Computation of integral bases in certain \(S_ n\) extensions of \({\mathbb{Q}}\) (Q1096672)

Let L be the decomposition field of an irreducible polynomial \(f\in {\mathbb{Z}}[X]\) with Galois group \(S_ n\). Let K be the fixed quadratic field of \(A_ n\subset S_ n\). The author proves that discriminant \(disc(f)=disc(K)\) if and only if \({\mathbb{Z}}[\alpha_ 1,...,\alpha_ n]\) is the ring of integers in L and L/K is unramified, where \(\alpha_ 1,...,\alpha_ n\) are the roots of f in L. Examples for \(f=x^ n+ax^ m+b\) are given. The special case with disc(f) squarefree was also dealt with by \textit{J. Elstrodt}, \textit{F. Grunewald} and \textit{J. Mennicke} in Glasgow Math. J. 27, 31-37 (1985; Zbl 0579.12005).