Real Algebra

Abstract

In the first section, we prove the Artin-Lang homomorphism theorem as a consequence of the Tarski-Seidenberg principle. We then present the real Nullstellensatz, which characterizes the ideals of polynomials vanishing on an algebraic set, when the ground field is real closed. The next two sections develop an ℜArtin-Schreier theoryℝ for rings, which is an extension of the theory presented in Chap. 1 for fields. The notion of a prime cone of a ring, which is related to that of a ring homomorphism into a real closed field, plays an important role and we shall encounter it later on in the study of the real spectrum in Chap. 7. In Section 4 we use the Artin-Schreier theory for rings, together with the Artin-Lang homomorphism theorem, to establish various versions of the Positivstellensatz, which characterizes polynomials which are positive on certain semi-algebraic subsets of R ⁿ. The last section presents the following criterion: an irreducible polynomial f ∈ R[X ₁, ..., X _n ] generates the ideal of polynomials vanishing on Z(f) if and only if f changes sign on R ⁿ.

Throughout this chapter R is a real closed field.