Galois groups of \(p\)-extensions with two ramification points
Let \(p\) be an odd prime and \(q\) a prime with \(q\equiv 1\pmod {p^{r+1}}\), \(q\not\equiv 1\pmod{p^{r+2}}\), \(r>0\). The author studies the Galois group \(G\) of the maximal \(p\)-extension \(K\) of \(\mathbb{Q}\) unramified outside \(p\) and \(q\). He shows that \(G\) modulo fourth commutators is generated by two elements \(\sigma,\tau\) which are connected by one generating relation \(\sigma^{p^{r+1}} \equiv[\tau^{p^r} [\tau,\sigma]^a, \sigma]\). He gives an implicit determination of \(a\) by means of a unique subfield of \(K\). Furthermore, he determines \(a\) explicitly in the cases \(p=3\), \(q\in\{19,37\}\).
scientific article; zbMATH DE number 1551510