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Generation of the ring of integers of the odd ray class fields of \(\mathbb Q(i)\) and division points of \(y^2=x^2-x\). (Génération de l’anneau des entiers des corps de classes de \(\mathbb Q(i)\) de rayon impair et points de division de \(y^2=x^2-x\).) (French) Zbl 0661.12003

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 \(\mathbb Z[i]\).

MSC:

11R33 Integral representations related to algebraic numbers; Galois module structure of rings of integers
11R11 Quadratic extensions

Citations:

Zbl 0647.12002
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References:

[1] Cassou-Noguès, Ph.; Taylor, M.J., (), Progress in Mathematics · Zbl 0935.19001
[2] Cassou-Noguès, Ph.; Taylor, M.J., A note on elliptic curves and the monogeneity of rings of integers, J. London math. soc., 37, 2, 63-72, (1988) · Zbl 0639.12001
[3] Cougnard, J., Conditions nécessaires de monogénéité. application aux extensions cycliques de degré, l ≥ 5 d’un corps quadratique imaginaire, J. London math. soc., 37, 2, 73-87, (1988) · Zbl 0647.12002
[4] Lang, S., ()
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