Lectures on Formally Real Fields
Prestel, Alexander.
Lectures on Formally Real Fields [electronic resource] / by Alexander Prestel. - XI, 128 p. online resource. - Lecture Notes in Mathematics, 1093 0075-8434 ; . - Lecture Notes in Mathematics, 1093 .
Orderings and semiorderings of fields -- Quadratic forms over formally real fields -- Real algebraic closures -- Some notions from model theory -- The transfer-principle for real closed fields -- The space of orderings and semiorderings -- Real places -- Real henselian fields -- SAP-fields -- Quadratic forms over formally real fields.
Absolute values and their completions - like the p-adic number fields- play an important role in number theory. Krull's generalization of absolute values to valuations made applications in other branches of mathematics, such as algebraic geometry, possible. In valuation theory, the notion of a completion has to be replaced by that of the so-called Henselization. In this book, the theory of valuations as well as of Henselizations is developed. The presentation is based on the knowledge aquired in a standard graduate course in algebra. The last chapter presents three applications of the general theory -as to Artin's Conjecture on the p-adic number fields- that could not be obtained by the use of absolute values only.
9783540390930
10.1007/BFb0101548 doi
Algebra.
Logic, Symbolic and mathematical.
Algebra.
Mathematical Logic and Foundations.
QA150-272
512
Lectures on Formally Real Fields [electronic resource] / by Alexander Prestel. - XI, 128 p. online resource. - Lecture Notes in Mathematics, 1093 0075-8434 ; . - Lecture Notes in Mathematics, 1093 .
Orderings and semiorderings of fields -- Quadratic forms over formally real fields -- Real algebraic closures -- Some notions from model theory -- The transfer-principle for real closed fields -- The space of orderings and semiorderings -- Real places -- Real henselian fields -- SAP-fields -- Quadratic forms over formally real fields.
Absolute values and their completions - like the p-adic number fields- play an important role in number theory. Krull's generalization of absolute values to valuations made applications in other branches of mathematics, such as algebraic geometry, possible. In valuation theory, the notion of a completion has to be replaced by that of the so-called Henselization. In this book, the theory of valuations as well as of Henselizations is developed. The presentation is based on the knowledge aquired in a standard graduate course in algebra. The last chapter presents three applications of the general theory -as to Artin's Conjecture on the p-adic number fields- that could not be obtained by the use of absolute values only.
9783540390930
10.1007/BFb0101548 doi
Algebra.
Logic, Symbolic and mathematical.
Algebra.
Mathematical Logic and Foundations.
QA150-272
512