Some algebra

Abstract

We want to study the locus V of roots of a finite set f_i(x₁,...,x_n) in kⁿ, (k being an algebraically closed the basic tool in this study is the ring of functions obtained by restricting polynomials from kⁿ to V. And we cannot get very far without knowing something about the algebra of such a ring. The purpose of this section is to prove 2 basic theorems from commutative algebra that are key tools in analyzing these rings, and hence also the loci such as V. We include these results because of their geometric meaning, which will emerge gradually in this chapter (cf. §7). On the other hand, we assume known the following topics in algebra:

• 1) The essentials of field theory (Galois theory, separability, transcendence degree).

• 2) Localization of a ring, the behaviour of ideals in localization, the concept of a local ring.

• 3) Noetherian rings, and the decomposition theorem of ideals in these rings.

• 4) The concept of integral dependence, (cf., for example, Zariski-Samuel, vol. 1).