Galois structure in weakly ramified extensions of \(\mathbb Q\)
This paper provides a freeness result for a suitable Galois module attached to so-called weakly ramified Galois extensions \(N\) of the rationals. We recall here that for tamely ramified extensions \(N/K\) the obvious candidate is the Galois module \(O_N\), but \(O_N\) is not locally free over \({\mathbb Z}[G]\) if \(N/K\) is not tame. It turns out however that there is a nice substitute for \(O_N\) in case \(N/K\) is weakly ramified of odd degree, to wit: the square root \(A_{N/K}\) of the inverse different [\textit{B. Erez}, Math. Z. 208, 239-255 (1991; Zbl 0737.11032)]. We have to explain terminology now: As is well known, \(N/K\) is tamely ramified at a prime \(\mathbf p\) iff the ramification exponent \(e_{\mathbf p}\) is prime to \(\mathbf p\), and \(N/K\) is wildly ramified if and only if it is not tame. Moreover, \(N/K\) is declared to be weakly ramified at \(\mathbf p\) if the second ramification group \(G_2({\mathbf p})\) is trivial. Theorem 1.1 gives a neat classification of abelian wildly and weakly ramified extensions of \({\mathbb Q}_p\): they are all gotten as subfields of the compositum of \(L\) and an unramified extension of \({\mathbb Q}_p\), where \(L\) is the degree \(p\) subfield of \({\mathbb Q}_p(\zeta_{p^2})\). At first glance this might be surprising because this forces the ramification exponent to be \(p\) (totally wild ramification); it cannot be \(p(p-1)\) for example, and indeed as the author points out, \({\mathbb Q}_p(\zeta_{p^2})\) is not weakly ramified over \({\mathbb Q}_p\). This seems to indicate that the class of weakly ramified extensions is in some way not very large. The main result of the paper (Theorem 1.2) states that if \(N/\mathbb Q\) is any weakly ramified Galois of odd degree and if all decomposition groups at wild places are abelian, then the locally free \({\mathbb Z}[G]\)-module \(A_{N}\) is free. The proof uses the Hom description of the class group, Gauss sum and resolvents, plus a fair amount of explicit calculation, aided by Theorem 1.1. In the concluding section, two numerical examples \(N\) with Galois group the nonabelian group \(G\) of order 27 and exponent 9 are given. The first satisfies all hypotheses of the theorem, and the second all but the abelianness condition. Nevertheless the author obtained by calculation (PARI) that \(A_{N}\) is free for the second example as well, and he suggests this might even be always the case for this choice of the Galois group.
scientific article; zbMATH DE number 1787790
scientific article