Relative integral bases of number fields of type \((q^ s,q^ s,\dots ,q^ s)\) (Q1309133)

Let \(K\) be an abelian number field. The author gives a description for the genus field of \(K\) and uses it to determine the conductor and discriminant of \(K\). The author also announces theorems which state that if \(L\) and \(K\) are number fields of type \((q^ s, q^ s, \dots, q^ s)\) with \(K\subseteq L\) and \(q\) a prime then \(L/K\) always has a relative integral basis when \(q\) is odd. Moreover, when \(q=2\), \(L/K\) has a relative integral basis provided \([L:K]\) is sufficiently large.