We present a new method for computing the Igusa class polynomials of a primitive quartic CM field. For a primitive quartic CM field, K, we compute the Igusa class polynomials modulo p for certain small primes p and then use the Chinese remainder theorem and a bound on the denominators to construct the class polynomials. We also provide an algorithm for determining endomorphism rings of Jacobian...

41 Elementary Number Theory 92 41.1 Subgroups of the Integers . . . . . . . . . . . . . . . . . . . . 92 41.2 Greatest Common Divisors . . . . . . . . . . . . . . . . . . . . 92 41.3 The Euclidean Algorithm . . . . . . . . . . . . . . . . . . . . . 93 41.4 Prime Numbers . . . . . . . . . . . . . . . . . . . . . . . . . . 96 41.5 The Fundamental Theorem of Arithmetic . . . . . . . . . . . . 97 4...

Let R be a (nonzero commutative unital) ring. If I and J are ideals of R such that R/I ⊕R/J is a cyclic R-module, then I + J = R. The rings R such that R/I ⊕R/J is a cyclic R-module for all distinct nonzero proper ideals I and J of R are the following three types of principal ideal rings: fields, rings isomorphic to K ×L for the fields K and L, and special principal ideal rings (R,M) such thatM...