Relative integral bases over a Hilbert class field
Let \(L/K\) be a Galois extension of number fields with \(L\supset H\), the Hilbert class field of \(K\). Then there exists a relative integral basis of \(L/H\) (i.e., \({\mathfrak o}_L\) is a free \({\mathfrak o}_H\)-module) if \(H\) has odd class number or if \([L:H]\) is odd or if \(G_{L/H}\) does not have cyclic Sylow 2-subgroups. In the exceptional case the existence is equivalent with the existence of a relative integral basis of \(H(\sqrt{\Delta})/H\), where \(\Delta\) is the discriminant of a field basis of \(L/H\). One knows [see \textit{E. Artin}, Collected Papers, Addison Wesley (1965; Zbl 0146.00101), repr. Springer (1982; Zbl 0493.01038), pp. 229-231] that the discriminant of \(L/H\) splits as a product \(d_{L/H}= \Delta\cdot {\mathfrak b}^2\), with \({\mathfrak b}\subset H\) a fractional ideal, and that the existence of a relative integral basis amounts to \({\mathfrak b}\) being principal. The key ingredient is then that an ideal of \(H\), which is invariant under the action of \(G_{H/K}\), is principal.
scientific article; zbMATH DE number 1891438