### Abstract:

It is often taken it for granted that all positive whole numbers except 0 and 1 can be factored
uniquely into primes. However, if K is a fi nite extension of the rational numbers, and OK
its ring of integers, it is not always the case that non-zero, non-unit elements of OK factor
uniquely. We do find, though, that the proper ideals of OK do always factor uniquely into
prime ideals. This result allows us to extend many properties of the integers to these rings.
If we a finite extension L of K and OL of OK, we find that prime ideals of OK need not remain prime when they are extended into OL; instead, they can split into a product of
prime ideals of OL in a very structured way. If L is a normal extension of K, we can use
Galois theory to further study this splitting by considering the intermediate fields of K
and L, as well as quotient rings of the associated rings of integers. In this paper, we will
introduce these topics of algebraic number theory, prove that unique factorization of ideals
holds using two different methods, and observe the patterns that arise in the splitting of
prime ideals.