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Valuation Theory

In: Basic Algebra

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  • P. M. Cohn

    (University College London, Department of Mathematics)

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.

Suggested Citation

  • P. M. Cohn, 2003. "Valuation Theory," Springer Books, in: Basic Algebra, chapter 9, pages 307-345, Springer.
    • Handle: RePEc:spr:sprchp:978-0-85729-428-9_9
    DOI: 10.1007/978-0-85729-428-9_9
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