Spaces of orderings and abstract real spectra.

*(English)*Zbl 0866.12001
Lecture Notes in Mathematics. 1636. Berlin: Springer. vi, 190 p. (1996).

This book is a comprehensive and self-contained introduction into the theory of spaces of orderings and abstract real spectra. It starts on a level suitable for a beginner, but the reader is expected to know elementary facts about ordered fields and valuations, and elementary commutative ring theory.

Chapter I makes the reader acquainted with the theory of orderings on fields and the connections to valuation theory.

Chapters II, III and IV introduce the reduced theory of quadratic forms and spaces of orderings, summing up all the important aspects and facts of this theory.

The rest of the book (Chapters V–VIII) is dedicated to the theory of abstract real spectra, starting out with the real spectrum of a ring. The author considers various sets of axioms defining abstract real spectra and proves that they are in fact equivalent, which is highly non-trivial, and are among the main contributions of these notes.

Chapter I makes the reader acquainted with the theory of orderings on fields and the connections to valuation theory.

Chapters II, III and IV introduce the reduced theory of quadratic forms and spaces of orderings, summing up all the important aspects and facts of this theory.

The rest of the book (Chapters V–VIII) is dedicated to the theory of abstract real spectra, starting out with the real spectrum of a ring. The author considers various sets of axioms defining abstract real spectra and proves that they are in fact equivalent, which is highly non-trivial, and are among the main contributions of these notes.

Reviewer: A.Tschimmel-Faltings (Bonn)

##### MSC:

12D15 | Fields related with sums of squares (formally real fields, Pythagorean fields, etc.) |

12-02 | Research exposition (monographs, survey articles) pertaining to field theory |

11E81 | Algebraic theory of quadratic forms; Witt groups and rings |

12J20 | General valuation theory for fields |

13J30 | Real algebra |

14P05 | Real algebraic sets |

14P10 | Semialgebraic sets and related spaces |

13A18 | Valuations and their generalizations for commutative rings |