Abstract
AbstractA detailed introduction to valuation theory of fields is given in this chapter. Valuations are presented from three different points of view: valuation maps, valuation rings, and places. Valuations are naturally associated with orderings, since every convex subring of an ordered field is a valuation ring. The precise relationship between the orderings of a field that are compatible with a valuation ring, and the orderings of the residue field of the latter, is exhibited by the Baer–Krull Theorem. Finally, the Artin–Lang Theorem is presented in the form of a Stellensatz. Other than the Artin–Schreier concept of orderings and real closures, this is the essential input for the solution of Hilbert’s 17th Problem, presented at the end of the chapter.
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