Steinitz classes of order 2 in quadratic and quartic fields (Q811365)

Let \(M/K\) be an extension of number fields, with relative discriminant \(\Delta_{M/K}\). Let \(d\) be the discriminant of any \(K\)-basis of \(M\), then \(\Delta_{M/K}=B^ 2(d)\) for some ideal \(B\) of \(K\). The ideal class of \(B\) is the Steinitz class of \(M/K\). The ideal classes of \(K\) which are Steinitz classes for normal extensions of odd prime degree \(\ell\) form a subgroup \(S\) of the ideal class group of \(K\). If the Steinitz class of \(M/K\) has order 2, then \(\Delta_{M/K}\) is principal but \(M\) has no relative integral basis over \(K\). Thus it is of interest to know if \[ (*)_ \ell \qquad S\text{ has an element of order }2. \] For \(K=\mathbb{Q}(\sqrt{d})\) the authors prove that if \(\ell\equiv 3\pmod 4\) then \(K\) has \((*)_ \ell\) iff the class number of \(K\) is even, recovering a result of \textit{S. Pierce} [Proc. Am. Math. Soc. 43, 39--41 (1974; Zbl 0287.12008)], and obtain results when \(\ell\equiv 5\pmod 8\). They also consider the case that \(K\) is a normal quartic extension of \(\mathbb{Q}\), obtaining the same result for \(\ell\equiv 3\pmod 4\) when \(K/\mathbb{Q}\) is cyclic. Finally, in the case \(K/\mathbb{Q}\) normal with Galois group \(C_ 2\times C_ 2\), they correct a result of Pierce [op. cit.].