New relative extension examples without normal basis
The setup is the following: All fields are supposed abelian over \(\mathbb{Q}\); \(N\) is the disjoint compositum of two other fields \(K\) and \(k\), \(K\) is tame over \(\mathbb{Q}\) with Galois group \(H\) of odd prime order \(l\), and the question is whether \(N/k\) has NIB or at least WNIB, where NIB means ``normal integral basis'', i.e. \(O_N\) is free as an \(O_k[H]\)-module; and WNIB (weak NIB) means that \({\mathcal M}O_N\) is free over the maximal order \({\mathcal M}\) of \(O_k[H]\). In the case \(N={\mathbb{Q}}(\zeta_p)\) it was shown by \textit{J. Brinkhuis} [J. Reine Angw. Math. 375/376, 157-166 (1987; Zbl 0609.12009)] and by \textit{J. Cougnard} [J. Number Theory 23, 336-346 (1986; Zbl 0588.12003)] that \(N/k\) does not have NIB. (Note that the above assumptions imply that \(l\) divides \(p-1\) exactly once.) In the present paper, more choices are allowed for \(k\), with \(K\) still the degree \(l\) field of prime conductor \(p\), and it is proved that most of the time \(N/k\) does not even have WNIB. The exact conditions on \(k\) are: \(k\) is cyclic of degree \(n\) over the rationals; \(p\) is totally ramified in \(k\), and the non-\(p\)-part \(n_0\) of \(n\) is not divisible by \(l\). The exact result is: If then \(N/k\) has WNIB, then \(k\) is \(\mathbb{Q}(\sqrt{p})\) or \({\mathbb{Q}}(\sqrt{-p})\); and even in this case, there is no NIB. The reviewer [ibid. 35, 180-193 (1990; Zbl 0718.11053)] showed some 10 years ago that one has WNIB rather often in this special case, but the question whether this is always so remains undecided. The author puts the established methods (resolvents in particular) to very good use. One main point is the fact that in the minus part the classes of totally ramified prime ideals tend to generate a part of the class group that is as large as possible. There is a very cleverly contrived example on p. 499 where one can explicitly ``see'' the nonexistence of a WNIB. A few small comments: The initial of Brinkhuis in the abstract should be {J}. \ On p. 495, the disjointness of \(N\) and \(\mathbb{Q}(\zeta_l)\) is already a consequence of the fact that \(p\) ramifies totally in \(N\) and not at all in \(\mathbb{Q}(\zeta_l)\). Perhaps the condition (H) can thus be avoided? Possibly there is an assumption in Corollary 4 which is not made explicit, i.e. that \({\mathcal P}O_{k^{(l)}}\) is principal. (If the latter condition does not hold, we are done anyway by Corollary 3.) The italic Q's on p. 501 seem to denote the rationals, which are also denoted by the usual \(\mathbb{Q}\) symbol in the paper. This paper considerably strengthens previous results. The assumption that \(N/k\) has prime degree might seem restrictive at first glance, but it isn't really, since the nonexistence results automatically carry over to any larger tame extension \(N'/k\). The condition that the degrees of \(K\) and \(k\) are relatively prime remains a certain restriction.
scientific article; zbMATH DE number 1796111