“p-adic Number”的概念、定义、习题解题方法及翻译-数学辞典

单词

p-adic Number

释义

p-adic Number

A -adic number is an extension of the Field of Rational Numbers such thatCongruences Modulo Powers of a fixed Prime are related to proximity inthe 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 aunique Integer. Then define the

-adic absolute value of

(2)

Also define the

-adic value

(3)

As an example, consider the Fraction

(4)

It has

-adic absolute values given by

			(5)
			(6)
			(7)
			(8)
			(9)

The -adic absolute value satisfies the relations

: 1. for all ,
: 2. Iff ,
: 3. for all and ,
: 4. for all and (the Triangle Inequality), and
: 5. for all and (the Strong Triangle Inequality).

In the above, relation 4 follows trivially from relation 5, but relations 4 and 5 are relevant in the moregeneral Valuation Theory.

The -adics were probably first introduced by Hensel in 1902 in a paper which was concerned with the development of algebraicnumbers in Power Series. -adic numbers were then generalized to Valuations by Kürschák in1913. In the early 1920s, Hasse formulated the Local-Global Principle (now usually called the Hasse Principle),which is one of the chief applications of Local Field theory. Skolem's -adic method, which is used in attackingcertain Diophantine Equations, is another powerful application of -adic numbers. Anotherapplication 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 zeroterms can always be added at the beginning)

(10)

with

an Integer,

the Integers between 0 and

inclusive, and where the sum isconvergent with respect to

-adic valuation. If

and

, then the expansion is unique. Burger andStruppeck (1996) show that for

a Prime and

a Positive Integer,

(11)

where the

-adic expansion of

(12)

and

(13)

For sufficiently large

(14)

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

(15)

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 beconstructed, and the set of

-adic numbers

is defined to be this completed space.

Just as the Real Numbers are the completion of the Rationals withrespect 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 thevaluation of both sides). Because the 2-adic numbers contain the rationals as a subset, we can immediately see that theequation 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 solutionsin , we show that it has no solutions in a Field Extension. For another example, consider . Thisequation 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 . Can we conclude that the equation has a solution in ? Unfortunately, in general, the answer is no, but thereare 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, Harmonic Number, Hasse Principle,Local Field, Local-Global Principle, Mahler-Lech Theorem, Product Formula, Valuation,Valuation Theory, von Staudt-Clausen Theorem

References

P-adic Numbers

Burger, E. B. and Struppeck, T. ``Does Really Converge? Infinite Series and -adic Analysis.'' Amer. Math. Monthly 103, 565-577, 1996.

Cassels, J. W. S. and Scott, J. W. Local Fields. Cambridge, England: Cambridge University Press, 1986.

Gouvêa, F. Q. -adic Numbers: An Introduction, 2nd ed. New York: Springer-Verlag, 1997.

Koblitz, N. -adic Numbers, -adic Analysis, and Zeta-Functions, 2nd ed. New York: Springer-Verlag, 1984.

Mahler, K. -adic Numbers and Their Functions, 2nd ed. Cambridge, England: Cambridge University Press, 1981.

随便看

数学辞典收录了8975条数学词条，基本涵盖了常用数学知识及数学英语单词词组的翻译及用法，是数学学习的有利工具。