Factoring Primes

First we will learn how, if $p\in\mathbf{Z}$ is a prime and $\O _K$ is the ring of integers of a number field, to write $p\O _K$ as a product of primes of $\O _K$. Then I will sketch the main results and definitions that we will study in detail during the next few chapters. We will cover discriminants and norms of ideals, define the class group of $\O _K$ and prove that it is finite and computable, and define the group of units of $\O _K$, determine its structure, and prove that it is also computable.