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.