Quadratic Forms over Real Fields
In this chapter we give a brief introduction to the theory of quadratic forms over a field K, and to its Wittring W(K), emphasizing the case where K is real. One reason for doing so is that the Zariski spectrum of W(K) reflects the collection of orderings of K. The main reason, however, is that Pfister’s Local-Glocal Principle (3.3.11) applied to the rational function field K = ℝ(X 1,...,X n ) gives a natural generalization of Hilbert’s 17th problem in Section 3.5.
KeywordsQuadratic Form Prime Ideal Positive Cone Real Field Transcendence Degree
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