We examine the factorization of algebraic integers in finite extensions of the rationals, especially imaginary quadratic extensions. We then look at unique factorization within ideal class groups of these extensions, and give an algorithm to compute the isomorphism class of the ideal class group. We also look at a formula for directly computing the class number for imaginary quadratic extensions of the rationals, and give complete lists of the extensions with a given class number n for odd n ≤ 23, and even n ≤ 6. We then revisit the factorization problem from the perspective of block monoids, which leads us to the Davenport constant and its use within block monoids and factorization in extensions of the rationals.
