7 - Determinant Theory

Summary

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.