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In this chapter, we recall basic concepts of p-adic theory: p-adic valuation, p-adic metrics, p-adic integers, and the likes. Most results are given without proofs since proofs can be found in a number of books on p-adic analysis (see, e.g., Mahler (p-adic Numbers and their Functions, 2nd edn., Cambridge Univ. Press (1981)), Schikhof (Ultrametric Calculus: An Introduction to p-adic Analysis, Cambridge Univ. Press (1984)), Koblitz (p-adic Numbers, p-adic Analysis, and Zeta-functions, Graduate texts in math., vol. 58, 2nd edn., Springer (1984)), Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Kluwer Academic Publishers, Dordrecht (1997), van Rooij, Non-Archimedean Functional Analysis, Marcel Dekker, New York (1978), Gouvêa, p-adic Numbers. An Introduction, 2nd edn., Springer, Berlin–Heidelberg–New York (1997), Robert, A Course in p-adic Analysis, Springer, Berlin–New York–Heidelberg (2000)), and, especially, Katok (p-adic Analysis in Comparison with Real, Mass. Selecta., American Mathematical Society (2003)).

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Rings and Fields of p-Adic Numbers

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  • 发表日期：

    2025年

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  • Vladimir Anashin

摘要

In this chapter, we recall basic concepts of p-adic theory: p-adic valuation, p-adic metrics, p-adic integers, and the likes. Most results are given without proofs since proofs can be found in a number of books on p-adic analysis (see, e.g., Mahler (p-adic Numbers and their Functions, 2nd edn., Cambridge Univ. Press (1981)), Schikhof (Ultrametric Calculus: An Introduction to p-adic Analysis, Cambridge Univ. Press (1984)), Koblitz (p-adic Numbers, p-adic Analysis, and Zeta-functions, Graduate texts in math., vol. 58, 2nd edn., Springer (1984)), Khrennikov, Non-Archimedean Analysis: Quantum Paradoxes, Dynamical Systems and Biological Models, Kluwer Academic Publishers, Dordrecht (1997), van Rooij, Non-Archimedean Functional Analysis, Marcel Dekker, New York (1978), Gouvêa, p-adic Numbers. An Introduction, 2nd edn., Springer, Berlin–Heidelberg–New York (1997), Robert, A Course in p-adic Analysis, Springer, Berlin–New York–Heidelberg (2000)), and, especially, Katok (p-adic Analysis in Comparison with Real, Mass. Selecta., American Mathematical Society (2003)).