Existence of integral bases for relative extensions of \(n\)-cyclic number fields
Sufficient conditions are given for an abelian extension \(L/K\) to have a relative integral basis when \(\text{Gal}(L/\mathbb{Q})\cong (\mathbb{Z}/q^s\mathbb{Z})^n\) and \(\text{Gal}(K/\mathbb{Q})\cong (\mathbb{Z}/q^r\mathbb{Z})^m\). Here, \(L=K_1K_2\dots K_n\) where each \(K_i/\mathbb{Q}\) is a cyclic extension of degree \(q^s\). For any prime \(p\), and an abelian extension \(F/\mathbb{Q}\), let \(e(p,F)\) denote the ramification index of \(p\) in \(F\) and \(q^{e_p}=\max_{l\leq i\leq n}(p,K_i)\). Let \([L:K]=q^M\). If \(q\) is an odd prime and \(M\geq e_p\) for all prime numbers \(p\) with \(e(p,L)\neq e(p,K)\) then \(L/K\) has a relative integral basis. For \(q=2\), a similar, but more complicated condition is given for the existence of a relative integral basis.
scientific article; zbMATH DE number 955267