## 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

### Keywords:

integers of ray class fields

Zbl 0647.12002
Full Text:

### 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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