Generation of the ring of integers of the odd ray class fields of \(\mathbb Q(i)\) and division points of \(y^2=x^2-x\)
type: Item
instance of: scholarly article
title: Generation of the ring of integers of the odd ray class fields of \(\mathbb Q(i)\) and division points of \(y^2=x^2-x\)
zbMATH Open document ID: 0661.12003
DOI: 10.1016/0022-314X(88)90013-3
author: Jean Cougnard
published in: Journal of Number Theory
publication date: 1988-01-01T00:00:00Z
review text: The rings of integers of ray class fields of \(\mathbb Q(i)\) having conductors relatively prime to 2 are shown to have a power basis over \(\mathbb Z[i]\). Let \(K\) be such a field and \(\mathbb Z_ K\) denote its ring of integers. It is shown that \(\mathbb Z_ K=\mathbb Z[i][\theta]\) where \(\theta =(T(\alpha)^ 2-1-2i)/4\). Here \(T(z)=\wp (1/2)/\wp (z)\) where \(\wp (z)\) is the Weierstrass \(\wp\)-function for \(\mathbb Z[i]\) and \(\alpha\) is a primitive point of \(\mathfrak f\) division of the elliptic curve \(\mathbb C/\mathbb Z[i]\) where \(\mathfrak f\) is the conductor of \(K\). An algorithm is given for determining an irreducible polynomial for \(\theta\) over \(\mathbb Q(i).\) Using recent results from another article of the author [J. Lond. Math. Soc., II. Ser. 37, No. 1, 73--87 (1988; Zbl 0647.12002)], a complete characterization is given of all cyclic extensions of prime degree \(\ell \geq 5\) of \(\mathbb Q(i)\) whose rings of integers have a power basis over \(\mathbb Z[i]\).
reviewed by: Charles J. Parry
Mathematics Subject Classification ID: 11R33
Mathematics Subject Classification ID: 11R11
zbMATH DE Number: 4079566
zbMATH Keywords: integers of ray class fields
MaRDI profile type: Publication
cites work: Adams operations and fixed point theorems in \(K\)-theory of group algebras
cites work: A Note on Elliptic Curves and the Monogeneity of Rings of Integers
cites work: Conditions Necessaires de Monogeneite. Application Aux Extensions Cycliques de Degre Premier l ⩾ 5 D'un Corps Quadratique Imaginaire
cites work: Q4075566
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description: scientific article; zbMATH DE number 4079566