Normal integral bases and strict ray class groups modulo 4 (Q1399674)

To every tame Abelian extension \(K/F\) with Galois group \(G\) one can attach the class of the projective \(O_F[G]\)-module \(O_K\) in Pic\((O_F[G])\); this map is called the Picard map, and the class of \(O_K\) is trivial iff \(K/F\) has an integral normal basis. The set of \(G\)-extensions \(K/F\) carries a group structure (Hasse product) if one allows \(F\) to be a Galois algebra, not necessarily a field, and it seems to be known that the Picard map is not a homomorphism in general. This general statement is made very explicit and concrete in the short paper under review. It is shown that if either \(F\) has \(h_F>1\) or non-2-elementary narrow 4-ray class group, the following situation arises infinitely often: \(K\) is a noncyclic Abelian extension of \(F\) of degree 4, and among the three quadratic subfields \(K_1,K_2,K_3\) of \(K/F\), exactly one is without normal integral basis. (Note that the Hasse product of the three extensions \(K_i/F\) is the trivial quadratic extension of \(F\), whose image under the Picard map is again trivial.) This phenomenon, which does not seem to have been looked at before, likewise prevents the existence of a maximal 2-elementary extension of \(F\) with a normal integral basis. Using standard methods, the paper also gives some general and computational information on the issue when \(F\) has 2-elementary 4-ray class group.