Diophantine equations are polynomial equations for which integer (or sometimes rational) solutions are sought. The oldest examples date from ancient Greek times, and Diophantus in particular solved many such equations. His methods and the questions they raised inspired much of modern number theory, beginning with the work of Fermat and Euler. Euler, and later Gauss, introduced algebraic integers to solve Diophantine equations, implicitly or explicitly using "unique prime factorization" to do so.

Extending the "integer" concept to algebraic numbers suggests the more general algebraic concept of ring. Likewise the concept of rational number suggests the algebraic concept of field. In this chapter we look specifically at fields of algebraic numbers and how to define their "integers." This involves the study of polynomial rings and the corresponding concepts of "prime" polynomial and "congruence modulo a prime." Then we return to algebraic number fields and view them "relative to" their subfields, such as the fields of rational numbers. This is facilitated by ideas from linear algebra, such as basis and dimension.

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).