The remaining step on the road to unique prime ideal factorization is to define "prime ideal" itself. This involves a definition of "division" for ideals. If we also define "product" of ideals, then a prime ideal is one with a property originally discovered by Euclid: if a prime p divides a product ab, then p divides a or p divides b. We then have all the ingredients needed for the definition of Dedekind domain: a Noetherian, integrally closed ring in which all prime ideals are maximal. And the main theorem follows: in a Dedekind domain, each ideal is a unique product of prime ideals.

Viewing an algebraic number field as a vector space relative to a subfield, which was foreshadowed in Chapter 4, involves varying the field of "scalars" in the definition of vector space. This leads in turn to relative concepts of "basis" and "dimension" which must be taken into account in algebraic number theory. In this chapter we review linear algebra from the ground up, with an emphasis on the relative point of view. This brings some nonstandard results into the picture, such as the Dedekind product theorem and the representation of algebraic numbers by matrices.

In algebraic number theory the determinant plays a bigger role than in a typical undergraduate linear algebra course. In particular, its relationship to trace, norm, and characteristic polynomial is important. For this reason, we develop determinant theory from scratch in this chapter, using an axiomatic characterization of determinant due to Artin. Among other things, this quickly gives basis-independence of the characteristic polynomial, trace, and norm. With these foundations we can introduce the discriminant, which tests whether an n-tuple of vectors form a basis, and paves the way for integral bases studied in the next chapter.

In the ring of integers of an algebraic number field, the obvious idea of "prime" is unsatisfactory, because "unique prime factorization" sometimes fails. This led Kummer to postulate the existence of "ideal numbers" outside the field, among which are "ideal primes" that restore unique prime factorization. Dedekind found that "ideal numbers" could be modeled by certain sets of actual numbers that he called ideals. In this chapter we give some concrete examples of ideals, then develop basic ideal theory, first in general rings, then in rings satisfying the ascending chain condition (ACC). ACC was identified by Emmy Noether as a key property of the rings studied by Dedekind, and shown by him to enjoy unique prime ideal factorization.

Solving quadratic Diophantine equations amounts to finding the values taken by quadratic forms, a problem that can be fruitfully approached by finding the equivalents of a given form under change of variables. This approach was initiated by Lagrange and developed to a high level by Gauss. However, the way Gauss did it involved an apparently difficult operation called composition of forms, clarified only later by the concept of Abelian group.

We follow Euclid from the elementary idea of division with remainder to unique prime factorization in the natural numbers, by way of the Euclidean algorithm. We also glimpse some more general concepts - algebraic integers, rings, and fields - that throw more light on ordinary integers. In particular, we show how the Pell equation can be solved with the help of quadratic integers.

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.

Modules are like vector spaces, except that their "scalars" are merely from a ring rather than a field. Because of this, modules do not generally have bases. However, we escape the difficulties in the rings of algebraic integers in algebraic number fields, and we can find bases for them with the help of the discriminant. This leads to another property of the latter rings - being integrally closed. In the next chapter we will see that the property of being integrally closed, together with the Noetherian property, is needed to characterize the rings in which unique prime ideal factorization holds.

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.