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

Any nonzero rational number  can be represented by

(1)

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

(2)

Also define the -adic norm

(3)

The -adics were probably first introduced by Hensel (1897) in a paper which was concerned with the development of algebraic numbers in power series. -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 -adic method, which is used in attacking certain Diophantine equations, is another powerful application of -adic numbers. Another application is the theorem that the harmonic numbers  are never integers (except for ). A similar application is the proof of the von Staudt-Clausen theorem using the -adic valuation, although the technical details are somewhat difficult. Yet another application is provided by the Mahler-Lech theorem.

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

(4)

with  an integer,  the integers between 0 and  inclusive, and where the sum is convergent with respect to -adic valuation. If  and , then the expansion is unique. Burger and Struppeck (1996) show that for  a prime and  a positive integer,

(5)

where the -adic expansion of  is

(6)

and

(7)

For sufficiently large ,

(8)

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

(9)

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

Just as the real numbers are the completion of the rationals  with respect to the usual absolute valuation , the -adic numbers are the completion of  with respect to the -adic valuation . The -adic numbers are useful in solving Diophantine equations. For example, the equation  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 .

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 , we show that it has no solutions in an extension field. For another example, consider . This equation has no solutions in  because it has no solutions in the reals , and  is a subset of .

Now consider the converse. Suppose we have an equation that does have solutions in  and in all the  for every prime . Can we conclude that the equation has a solution in ? 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.

REFERENCES:

Burger, E. B. and Struppeck, T. "Does  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. "-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. https://arxiv.org/abs/math-ph/9911027.

Weisstein, E. W. "Books about P-adic Numbers." https://www.ericweisstein.com/encyclopedias/books/P-adicNumbers.html.

Wolfram, S. A New Kind of Science. Champaign, IL: Wolfram Media, p. 1168, 2002.

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