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Basic Algebra pp 307-345 | Cite as

Valuation Theory

  • P. M. Cohn

Abstract

Valuation theory may be described as the study of divisibility (in commutative rings) in its purest form, but that is only one aspect. The general formulation leads to the introduction of topological concepts like completion, which provides a powerful tool. It also emphasizes the parallel with the absolute value on the real and complex numbers. After the initial definitions (in Section 9.1) we shall prove the essential uniqueness of the absolute value on R and C (in Section 9.2) and go on to describe the p-adic numbers in Section 9.3 and integral elements in Section 9.4, before looking at simple cases of the extension problem in Section 9.5.

Keywords

Maximal Ideal Integral Domain Valuation Ring Irreducible Polynomial Principal Ideal Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Professor P.M. Cohn 2003

Authors and Affiliations

  • P. M. Cohn
    • 1
  1. 1.Department of MathematicsUniversity College LondonLondonUK

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