Sur la monogeneite de l'anneau des entiers de certains corps de rayon

The computation of sextic fields with a quadratic subfield

“…Assume first that F is quadratic over some subfield E. Since F is Galois over E, then F' /E is Galois if and only if H is invariant under &al(F/E), and, when this condition is fulfilled, coF>/F is a homomorphism of & al (F / E)-modules (cf., for instance, [14,Ch. 11,§3]).…”

“…Let us look more closely at the first three exceptions in the totally imaginary case, namely the fields with discriminants dK = (-8)3 • 625, (-19)3 • 49 (the ones referred to in the introduction), and (-20) • 49, for which we found f = 2, 5 and 3, respectively (actually the smallest possible values for /). That / must be greater than 1 comes from local reasons in the first case ( / must be even), from [3] in the second case, and from the fact that ZA is not a free Zk -module in the last case.…”

On power bases for rings of integers of relative Galois extensions

“…Since cyclotomic fields are ray class fields for Q and their rings of integers have power bases which are generated by roots of unity, it is natural to consider ray class fields of imaginary quadratic fields. More precisely, one considers for an integral ideal F of an imaginary quadratic field k, whether the ring of integers of the ray class field k F mod F has a power basis over that of the Hilbert class field of k. This does not hold in general (see [4]), but many affirmative results under some conditions have been obtained by considering division points of elliptic functions as generators (see [2,3,6,20]).…”