Generation of the ring of integers of the odd ray class fields of \(\mathbb Q(i)\) and division points of \(y^2=x^2-x\)
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 \(\mathbb Z[i]\).
scientific article; zbMATH DE number 4079566
scientific article