p-adic Number

A $p$-adic number is an extension of the Field of Rational Numbers 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

$$x={p^a r\over s},$$ (1)

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 absolute value of $x$ by

$$\vert x\vert _p=p^{-a}.$$ (2)

Also define the $p$-adic value

$$\vert\vert _p=0.$$ (3)

As an example, consider the Fraction

$${\textstyle{140\over 297}}=2^2\cdot 3^{-3}\cdot 5\cdot 7\cdot 11^{-1}.$$ (4)

It has $p$-adic absolute values given by

$$\vert{\textstyle{140\over 297}}\vert _2 = {\textstyle{1\over 4}}$$ (5)

$$\vert{\textstyle{140\over 297}}\vert _3 = 27$$ (6)

$$\vert{\textstyle{140\over 297}}\vert _5 = {\textstyle{1\over 5}}$$ (7)

$$\vert{\textstyle{140\over 297}}\vert _7 = {\textstyle{1\over 7}}$$ (8)

$$\vert{\textstyle{140\over 297}}\vert _{11} = 11.$$ (9)

The $p$-adic absolute value satisfies the relations

1. $\vert x\vert _p\geq 0$ for all $x$,

2. $\vert x\vert _p=0$ Iff $x=0$,

3. $\vert xy\vert _p=\vert x\vert _p\,\vert y\vert _p$ for all $x$ and $y$,

4. $\vert x+y\vert _p\leq \vert x\vert _p+\vert y\vert _p$ for all $x$ and $y$ (the Triangle Inequality), and

5. $\vert x+y\vert _p\leq {\rm max}(\vert x\vert _p, \vert y\vert _p)$ for all $x$ and $y$ (the Strong Triangle Inequality).

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

The $p$-adics were probably first introduced by Hensel in 1902 in a paper which was concerned with the development of algebraic numbers in Power Series. $p$-adic numbers were then generalized to Valuations by Kürschák in 1913. 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 $p$-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)

$$x=\sum_{j=m}^\infty a_j p^j,$$ (10)

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 $x\not=0$ and $a_m\not=0$, then the expansion is unique. Burger and Struppeck (1996) show that for $p$ a Prime and $n$ a Positive Integer,

$$\vert n!\vert _p=p^{-(n-A_p(n))/(p-1)},$$ (11)

where the $p$-adic expansion of $n$ is

$$n=a_0+a_1p+a_2p^2+\ldots+a_LP^L,$$ (12)

and

$$A_p(n)=a_0+a_1+\ldots+a_L.$$ (13)

For sufficiently large $n$,

$$\vert n!\vert _p\leq p^{-n/(2p-2)}.$$ (14)

The $p$-adic valuation on $\Bbb{Q}$ gives rise to the $p$-adic metric

$$d(x,y) = \vert x-y\vert _p,$$ (15)

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 $\Bbb{Q}_p$ is defined to be this completed space.

Just as the Real Numbers are the completion of the Rationals $\Bbb{Q}$ with respect to the usual absolute valuation $\vert x-y\vert$, the $p$-adic numbers are the completion of $\Bbb{Q}$ with respect to the $p$-adic valuation $\vert x-y\vert _p$. 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 $\Bbb{Q}$, we show that it has no solutions in a Field Extension. For another example, consider $X^2 + 1 = 0$. This equation has no solutions in $\Bbb{Q}$ because it has no solutions in the reals $\Bbb{R}$, and $\Bbb{Q}$ is a subset of $\Bbb{R}$.

Now consider the converse. Suppose we have an equation that does have solutions in $\Bbb{R}$ and in all the $\Bbb{Q}_p$. Can we conclude that the equation has a solution in $\Bbb{Q}$? 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, 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 $\sum_{n=0}^\infty {1\over n!}$ Really Converge? Infinite Series and $p$-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. $P$-adic Numbers: An Introduction, 2nd ed. New York: Springer-Verlag, 1997.

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

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