A Short Review on p-Adic Numbers

Author :Mutlay Dogan1
1Department of Mathematics, Ishik University, Erbil, Iraq

Abstract:In this review work we indicated that there are two different metrics to find the distance between any two rational numbers. One of these metrics is usual absolute value  |.|  and the other one is the p-adic absolute value |.|p , here p is a prime number. Most crucial property of this norm is that it satisfies the ultra-metric triangle inequality. In this work we gave some definitions and properties about both metric and ultra metric norms. Especially, we reviewed the construction of p-adic number field. Rational numbers have two types of completions; while one of them is real numbers field the other one is p-adic numbers field.

Keywords: p-Adic Numbers, Normed Fields, Non-Archimedean Norm, Completion of Rational Number

doi: 10.23918/eajse.v3i2p121

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Arefeva, I.Y., Dragovich, B., Frampton, P.H., & Volovich, I.V. (1991). The Wave-Function of the
Universe and P-Adic Gravity. Int J Mod Phys A, 6, 4341-58.
Dogan, M. (2015). Cayley ağacı üzerinde p-adik Ising-Vannimenus ve p-adik
 -modellerinin padik Gibbs ölçümleri ve faz geçişleri, PhD Thesis.
Ganikhodjaev, N.N., Rozikov, U.A. (2009). On Ising Model with Four Competing Interactions on
Cayley Tree. Math Phys Anal Geom, 12, 141-56.
Katok, S. (2007). p-Adic Analysis Compared with Real. In. Eds. USA: American Mathematical
Khrennikov, A.Y. (1997). Non-Archimedean Analysis: Quantum paradoxes, Dynamical systems and
Biological models. In. Eds. Dordrecht: Kluwer Academic Publisher, p. 376.
Khrennikov, A.Y., Ludkovsky, S. (2003). Stochastic processes on non-Archimedean spaces with
values in non-Archimedean fields. Markov Process Relat, 9, 131-62.

Khrennikov, A.Y., Nilson, M. (2004). p-adic deterministic and random dynamical systems. In. Eds.
Dordreht: Kluwer, p. 269.
Koblitz, N.(1977). P-Adic Numbers, P-Adic Analysis And Zeta-Function. Berlin: Springer.
Robert, A.M. (2000). A Course of p-Adic Analysis. In. Eds. New York: Springer
Rozikov, U.A. (2013). Gibbs Measures on Cayley Trees. In. Eds. Singapur: World Scientific, p. 383.