0. The values of the modular function 7 (τ) at imaginary quadratic arguments τ in the upper half plane are known s singular moduli. They are all algebraic integers. In this paper we will study the prime factorizations of the differences of two singular moduli. These differences turn out to be highly divisible numbers. For instance, we will determine the set of primes dividing the absolute norm ofy'(T) — 1 728 =7 (τ) —7(1) (and the multiplicities with which they occur); they turn out to be contained among the prime divisors of the positive integers of the form d—x, where d is the discriminant of τ, and hence smaller than or equal to d, e.g. :

On singular moduli.

https://www.rzuser.uni-heidelberg.de/~hb3/publ/pcft.pdf

Class Field Towers

Let N be a non-abelian normal CM-eld of degree 4p; p any odd prime. Note that the Galois group of N is either the dicyclic group of order 4p; or the dihedral group of order 4p: We prove that the (relative) class number of a dicyclic CM-eld of degree 4p is always greater then one. Then, we determine all the dihedral CM-elds of degree 12 with class number one: there are exactly nine such CM-elds.

/pdf/the-class-number-one-problem-for-some-non-abelian-normal-cm-51kvh0paao.pdf

The class number one problem for some non-abelian normal CM-fields

A large class of algorithms for computing resolvents of algebraic equations — so called rational transformations — is investigated and characterized group theoretically. The concept of rational transformations implies a program how to develop good methods to determine the Galois group of an equation. It is shown that some known methods are special cases of rational transformations, and a new procedure to find the group of a sextic equation is given. Moreover, all cases in which Galois resolvents can be found by means of rational transformations are classified.

Noriko Yui

Papers

Polynomials with Dp as Galois group