Applications of Prime Factorization of Ideals in Number Fields

For a number fieldK, that is, a finite extension of Q, and a prime number p, a fundamental theorem of algebraic number theory implies that the ideal (p) ⊆ OK factors uniquely into prime ideals as (p) = p1 1 · · · p eg g . In this paper we explore different interpretations of this using the factorization of polynomials in finite and p-adic fields and Galois theory. In particular, we present some concrete applications for which these different interpretations are useful: • We use the Cebotarev Density Theorem to prove Dirichlet’s Theorem by computing the Frobenius element in a cyclotomic field in Section 3.1. • We prove quadratic reciprocity in Section 3.2 by using two different methods to determine how a prime splits in a particular number field. • We will also investigate in Section 3.3 under what conditions the existence of a rational root to a polynomial is guaranteed given a real root and a root in Qp for each prime. This is known as the Hasse Principle, and we will show that it holds for irreducible polynomials in one variable, and also construct a counterexample to show that it doesn’t hold for every polynomial in one variable.