Notes on Introductory Algebraic Number Theory

This paper introduces the basic results of Algebraic Number Theory. Accordingly, having established the existence of integral bases and the result that ideals in Dedekind domains can be uniquely decomposed into prime ideals, we then give the relation between ramification index, residue class degree and the degree of the extension. Moreover, we also demonstrate the connection between the decomposition group and the Galois groups of certain tower extensions. We then employ Minkowski’s bound to prove several properties of algebraic number fields. Furthermore, we develop this theory in the context of quadratic and cyclotomic extensions of Q in order to prove quadratic reciprocity and to demonstrate the strong relationship between the Čebotarev and Dirichlet prime density theorems. This paper assumes a background knowledge of Commutative Algebra and Galois theory.