p-adic Number

A p-adic number is an extension of the field of rationals such that congruences modulo powers of a fixed prime p are related to proximity in the so called "p-adic metric."

Any nonzero rational number x can be represented by


where p is a prime number, r and s are integers not divisible by p, and a is a unique integer. Then define the p-adic norm of x by


Also define the p-adic norm


The p-adics were probably first introduced by Hensel (1897) in a paper which was concerned with the development of algebraic numbers in power series. p-adic numbers were then generalized to valuations by Kűrschák in 1913. Hasse (1923) subsequently formulated the Hasse principle, which is one of the chief applications of local field theory. Skolem's p-adic method, which is used in attacking certain Diophantine equations, is another powerful application of p-adic numbers. Another application is the theorem that the harmonic numbers H_n are never integers (except for H_1). A similar application is the proof of the von Staudt-Clausen theorem using the p-adic valuation, although the technical details are somewhat difficult. Yet another application is provided by the Mahler-Lech theorem.

Every rational x has an "essentially" unique p-adic expansion ("essentially" since zero terms can always be added at the beginning)


with m an integer, a_j the integers between 0 and p-1 inclusive, and where the sum is convergent with respect to p-adic valuation. If x!=0 and a_m!=0, then the expansion is unique. Burger and Struppeck (1996) show that for p a prime and n a positive integer,


where the p-adic expansion of n is




For sufficiently large n,


The p-adic valuation on Q gives rise to the p-adic metric


which in turn gives rise to the p-adic topology. It can be shown that the rationals, together with the p-adic metric, do not form a complete metric space. The completion of this space can therefore be constructed, and the set of p-adic numbers Q_p is defined to be this completed space.

Just as the real numbers are the completion of the rationals Q with respect to the usual absolute valuation |x-y|, the p-adic numbers are the completion of Q with respect to the p-adic valuation |x-y|_p. The p-adic numbers are useful in solving Diophantine equations. For example, the equation X^2=2 can easily be shown to have no solutions in the field of 2-adic numbers (we simply take the valuation of both sides). Because the 2-adic numbers contain the rationals as a subset, we can immediately see that the equation has no solutions in the rationals. So we have an immediate proof of the irrationality of sqrt(2).

This is a common argument that is used in solving these types of equations: in order to show that an equation has no solutions in Q, we show that it has no solutions in an extension field. For another example, consider X^2+1=0. This equation has no solutions in Q because it has no solutions in the reals R, and Q is a subset of R.

Now consider the converse. Suppose we have an equation that does have solutions in R and in all the Q_p for every prime p. Can we conclude that the equation has a solution in Q? Unfortunately, in general, the answer is no, but there are classes of equations for which the answer is yes. Such equations are said to satisfy the Hasse principle.

See also

Ax-Kochen Isomorphism Theorem, Diophantine Equation, Greatest Dividing Exponent, Harmonic Number, Hasse Principle, Local Field, Mahler-Lech Theorem, p-adic Integer, p-adic Norm, Product Formula, Valuation, Valuation Theory, von Staudt-Clausen Theorem

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Burger, E. B. and Struppeck, T. "Does sum_(n=0)^(infty)1/(n!) Really Converge? Infinite Series and p-adic Analysis." Amer. Math. Monthly 103, 565-577, 1996.Cassels, J. W. S. Ch. 2 in Lectures on Elliptic Curves. New York: Cambridge University Press, 1991.Cassels, J. W. S. and Scott, J. W. Local Fields. Cambridge, England: Cambridge University Press, 1986.De Smedt, S. "p-adic Arithmetic." The Mathematica J. 9, 349-357, 2004.Gouvêa, F. Q. P-adic Numbers: An Introduction, 2nd ed. New York: Springer-Verlag, 1997.Hasse, H. "Über die Darstellbarkeit von Zahlen durch quadratische Formen im Körper der rationalen Zahlen." J. reine angew. Math. 152, 129-148, 1923.Hasse, H. "Die Normenresttheorie relativ-Abelscher Zahlkörper als Klassenkörpertheorie in Kleinen." J. reine angew. Math. 162, 145-154, 1930.Hensel, K. "Über eine neue Begründung der Theorie der algebraischen Zahlen." Jahresber. Deutsch. Math. Verein 6, 83-88, 1897.Kakol, J.; De Grande-De Kimpe, N.; and Perez-Garcia, C. (Eds.). p-adic Functional Analysis. New York: Dekker, 1999.Koblitz, N. P-adic Numbers, P-adic Analysis, and Zeta-Functions, 2nd ed. New York: Springer-Verlag, 1984.Koch, H. "Valuations." Ch. 4 in Number Theory: Algebraic Numbers and Functions. Providence, RI: Amer. Math. Soc., pp. 103-139, 2000.Mahler, K. P-adic Numbers and Their Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1981.Ostrowski, A. "Über sogennante perfekte Körper." J. reine angew. Math. 147, 191-204, 1917.Vladimirov, V. S. "Tables of Integrals of Complex-Valued Functions of p.-adic Arguments" 22 Nov 1999., E. W. "Books about P-adic Numbers.", S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1168, 2002.

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p-adic Number

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Weisstein, Eric W. "p-adic Number." From MathWorld--A Wolfram Web Resource.

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