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Recent results on the monogeneity of certain rings of integers. (Résultats récents sur la monogénéïté de certains anneaux d’entiers.) (French) Zbl 0712.11062

Sémin. Théor. Nombres, Univ. Bordeaux I 1987-1988, Exp. No. 32, 12 p. (1988).
Definition: Let \(L/K\) be an extension of an algebraic number field, \(Z_ L\) (resp. \(Z_ K\)) the ring of integers of \(L\) (resp. \(K\)). Then \(Z_ L\) is \(Z_ K\) monogenic if there exists \(\theta\) such that \(Z_ L=Z_ K[\theta]\). The author, first of all, gives some known conditions under which the following conjecture is true: Let \(F\) be an ideal of \(Z_ K\), \(H_ K\) the Hilbert class field of \(K\), \(K^{(F)}\) the class field of \(K\) of ray \(F\). Then the ring of integers of \(K^{(F)}\) is monogenic over that of \(H_ K.\)
Finally, the author proves the following result concerning cyclic extensions of \(K=\mathbb Q(\sqrt{-2})\): Let \(L\) be a cyclic extension of \(K\) of prime degree \(q\geq 5\). Then \(L\) has its ring of integers monogenic over \(\mathbb Z[\sqrt{-2}]\) if and only if one of the following two conditions is satisfied:
a) \(p=2q+1\) is prime and \(L\) is the composition of \(K\) and the real maximal subfield of the \(p\)-th cyclotomic field.
b) \(q\equiv 1\pmod 4\), \(p=2q+1\) is prime and \(L\) is the ray class field for one of the prime ideals of \(\mathbb Z_ K\) over \(p\).
The proof uses parametrisation of the elliptic curve \(E=C/\mathbb Z_ K\) introduced by Ph. Cassou-Noguès and M. J. Taylor [Sémin. Théor. Nombres, Paris, 1986–87, Prog. Math. 75, 35–64 (1988; Zbl 0714.11078)].
Reviewer: R.Ph.Steiner

MSC:

11R21 Other number fields
11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R37 Class field theory
11G05 Elliptic curves over global fields