## A note on elliptic curves and the monogeneity of rings of integers.(English)Zbl 0639.12001

Call an extension $$L| M$$ of number fields “monogenic” if $${\mathfrak O}_ L={\mathfrak O}_ M[x]$$ with some $$x\in {\mathfrak O}_ L$$. By results of M.-N. Gras and Cougnard, monogenic Galois extensions of M of prime degree $$p\geq 5$$ are rare if $$M={\mathbb{Q}}$$ or M imaginary quadratic, i.e., the possible extensions of this type are those constructed from certain class fields of M.
Let now K be imaginary quadratic with Hilbert class field H. The main result (theorem 1) of the article states that $$L| M$$ is indeed monogenic (a generator of $${\mathfrak O}_ L$$ over $${\mathfrak O}_ M$$ being constructed by means of elliptic functions) if M contains H, L is a certain class field obtained from K, and the whole situation is subject to certain technical assumptions which will not be repeated here. In theorems 2 and 3, the authors descend their result to a statement on the monogeneity of certain extensions over the Hilbert class field H.
As an application, they mention the following beautiful example: Let $$K={\mathbb{Q}}(\sqrt{-7})$$, $$\ell =2p+1=\bar {\mathfrak l}\cdot {\mathfrak l}^ a$$prime that splits in K. Then the ray class fields $$K({\mathfrak l})$$ and $$K(\bar{\mathfrak l})$$ are monogenic over K.
Reviewer: E.-U.Gekeler

### MSC:

 11R99 Algebraic number theory: global fields 14H52 Elliptic curves 11R37 Class field theory 14H25 Arithmetic ground fields for curves 14H45 Special algebraic curves and curves of low genus 11R11 Quadratic extensions
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