2 - Some Basics of Algebraic Number Theory

Summary

In Chapter 2, we explain some of the basics of algebraic number theory, which we will need in Chapter 3 to introduce the theory of heights and to give a proof of the Mordell-Weil theorem. We begin by introducing the trace and the norm of an element of a finite extention field. We show the existence of an integral basis for a ring of integers and define the discriminant of a number field. After showing the existence of a prime factorization of a fractional ideal of a ring of integers (Theorems 2.5 and 2.6), we prove Minkowski's convex body theorem (Theorem 2.9) and Minkowski's discriminant theorem (Theorem 2.13). Finally, we introduce the notions of the ramification index and the residue degree at a prime ideal of an extension field. We define the difference of a number field, and explain several results relating the discriminant, the difference, and the ramifications of prime ideals (Lemma 2.17 and Theorem 2.18).