Let be a field and let . Describe the ideals of in terms of the factorization of .
We begin with a lemma.
Lemma: Let be a principal ideal domain and let factor into irreducibles as . Then the ideals of have the form where and . Proof: is a unique factorization domain. By the lattice isomorphism theorem for rings, the ideals of correspond to the ideals of which contain , and these in turn correspond to the divisors of . In this previous exercise, we characterized the divisors of in terms of its factorization as desired.
For this problem, since is a field, is a Euclidean domain and thus a principal ideal domain.