P-adic Numbers

The ''p''-adic number systems were first described by Kurt Hensel in 1897 . For each Prime Number ''p'', the ''p''-adic Number System extends the ordinary Arithmetic of the Rational Numbers in a way different from the extension of the rational number system to the Real and Complex number systems. The main use of these other systems is in Number Theory .

The extension is achieved by an alternative interpretation of the concept of Absolute Value . The ''p''-adic numbers were motivated primarily by an attempt to bring the ideas and techniques of Power Series methods into number theory. Their influence now extends far beyond this. For example, the field of ''p''-adic Analysis essentially provides an alternative form of Calculus .

More formally, for a given prime ''p'', the Field Q_''p'' of ''p''-adic numbers is an Extension Field of the Rational Numbers . If all of the fields Q_''p'' are collectively considered, we arrive at Helmut Hasse 's Local-global Principle , which roughly states that certain equations can be solved over the rational numbers If And Only If they can be solved over the Real Numbers ''and'' over the ''p''-adic numbers for every prime ''p''.
(This principle is not always true; consider it more accurately as a motivating idea. In special cases, such as with quadratic forms or division algebras over global fields,
the principle can be proved in a precise form.)
The field Q_''p'' is also given a Topology derived from a Metric , which is itself derived from an alternative Valuation on the rational numbers. This metric is Complete in the sense that every Cauchy Sequence converges. This is what allows the development of calculus on Q_''p'', and it is the interaction of this analytic and algebraic structure which gives the ''p''-adic number systems their power and utility.

In situations where one is considering one special prime and then other primes enter the story, the first prime is often called ''p'' and the others
are called

\ell

, following the notation used by Jean-Pierre Serre . For instance, one might speak of the representation of
an inertia group at ''p'' on the

\ell

-adic Tate Module , where

\ell

does not equal ''p''.

MOTIVATION

The simplest introduction to ''p''-adic numbers is to consider 10-adic
integers, which are simply strings of digits in which you allow an infinite
number of digits to the left, for example, the number ...9999, and
then do arithmetic with such numbers as usual. In other words, do arithmetic
like you would with real numbers, but with digits going off to the left
instead of to the right. The references to
valuations and metrics given below are simply technical devices which
justify the ordinary operations. For example, one has the computation

:

which is true because there are an infinite number of carries which never
end, so there will never be a digit "1" on the left in the result. So a first
10-adic result is that ...999 = −1. It follows from this that negative
integers can be represented as digit expansions in which all
lefthand digits are eventually equal to 9. This is analogous to Two's Complement Notation in computer science, in which negative integers
are coded with the leftmost bit being set to 1: in the 2-adic integers, negative integers will correspond to numbers
in which all lefthand digits are eventually equal to 1 (in general, ''p'' − 1 for ''p''-adic
numbers).

From ...999 = −1 we can derive other 10-adic representations, such as

:

rac{-1}{3}=rac{...999}{3}=...333

rac{-2}{3}=2\left (rac{-1}{3} 
ight )=2(...333)=...666

rac{1}{3}=rac{-2}{3}+1=...666+1=...667

rac{2}{3}=rac{-1}{3}+1=...333+1=...334.

One point that confuses many people is why the ''p'' in ''p''-adic numbers is
always prime. As seen above, it is not absolutely necessary, as things work
well enough in base 10. (Often the term ''g-adic number'' is used when the concept is used for a fixed Composite Number ''g''.) However, ''p''-adic numbers are most useful for
doing calculus-type computations, and it is important to be
able to divide, that is, one wants to work in a Field . The
point is that ''p''-adic numbers form a field if and only if ''p'' is a
prime power, and you get the same result for a prime power as you do
for the prime (e.g., base 16 is just shorthand for base 2). In particular, if ''p'' is not a
prime power, then you can always find two
nonzero ''p''-adic numbers ''A'' and ''B'' such that ''AB'' = 0, which removes all possibility of finding their inverses. It is an interesting exercise
to find such numbers for ''p'' = 10, for example, the following (check
that the products are well defined over the 10-adics):

:

A = \prod_{n = 1}^\infty \, (2^{-1} {
m mod}\, 5^n),
\qquad
B = \prod_{n = 1}^\infty \, (5^{-1} {
m mod}\, 2^n) .

If ''p'' is a fixed prime number, then any Integer can be written as a ''p-adic expansion'' (writing the number in "base ''p''") in the form
:

\pm\sum_{i=0}^n a_i p^i

where the a_i are integers in {0,...,''p'' − 1}. This is expressed by saying that the integer has been "written in base ''p''". For example, the 2-adic or Binary expansion of 35 is 1·2⁵ + 0·2⁴ + 0·2³ + 0·2² + 1·2¹ + 1·2⁰, often written in the shorthand notation 100011₂.

The familiar approach to generalizing this description to the larger domain of the rationals (and, ultimately, to the reals) is to include sums of the form:

:

\pm\sum_{i=-\infty}^n a_i p^i

A definite meaning is given to these sums based on Cauchy Sequence s, using the Absolute Value as metric. Thus, for example, 1/3 can be expressed in base 5 as the limit of the sequence 0.1313131313...₅. In this formulation, the integers are precisely those numbers which can be represented in the form where ''a''_''i'' = 0 for all ''i'' < 0.

As an alternative, if we extend the ''p''-adic expansions by allowing infinite sums of the form

:

\sum_{i=k}^{\infty} a_i p^i

where ''k'' is some (not necessarily positive) integer, we obtain the Field Q_''p'' of '''''p''-adic numbers'''. Those ''p''-adic numbers for which ''a''_''i'' = 0 for all ''i'' < 0 are also called the '''''p''-adic integers'''. The ''p''-adic integers form a Subring of Q_''p'', denoted '''Z'''_''p''. (Note: '''Z'''_''p'' is often used to represent the set of integers modulo ''p''. If each set is needed, the latter is usually written '''Z'''/''p'''''Z''' or '''Z'''/''p''. Be sure to check the notation for any text you read.)

Intuitively, as opposed to ''p''-adic expansions which extend to the ''right'' as sums of ever smaller, increasingly negative powers of the base ''p'' (as is done for the real numbers as described above), these are numbers whose ''p''-adic expansion to the ''left'' are allowed to go on forever. For example, the ''p''-adic expansion of 1/3 in base 5 is ...1313132, i.e. the limit of the sequence 2, 32, 132, 3132, 13132, 313132, 1313132, ... Informally, we can see that multiplying this "infinite sum" by 3 in base 5 gives ...0000001. As there are no negative powers of 5 in this expansion of 1/3 (i.e. no numbers to the right of the decimal point), we see that 1/3 is a ''p''-adic integer in base 5.

The main technical problem is to define a proper notion of infinite sum which makes these expressions meaningful - this requires the introduction of the ''p''-adic Metric . Two different but equivalent solutions to this problem are presented below.

CONSTRUCTIONS

Analytic approach

The Real Number s can be defined as Equivalence Class es of Cauchy Sequence s of Rational Number s; this allows us to, for example, write 1 as 1.000... = 0.999... . However, the definition of a Cauchy sequence relies on the Metric chosen and, by choosing a different one, numbers other than the real numbers can be constructed. The usual metric which yields the real numbers is called the Euclidean Metric .

For a given prime ''p'', we define the ''p-adic metric'' in Q as follows:

:<math>x 2

2 \,\!</math>

:<math>x 3

1/9 \,\!</math>

:<math>x 5

25 \,\!</math>

:<math>x 7

1/7 \,\!</math>

:<math>x {11}

11 \,\!</math>

:<math>d P(x,y) x-y_p \,\!</math>
:''f'': '''Q'''''p'' &rarr '''Q'''''p'', ''f''(''x'') (1/''x''''p'')2 for ''x'' &ne 0, ''f''(0) = 0,
:<math>x P c^{-\operatorname{ord}_P(x)}</math>

	APPAREL
	BABY
	BEAUTY
	BOOKS
	CAR TOYS
	CELL PHONES
	DVD'S
	ELECTRONICS
	GOURMET FOOD
	GROCERIES
	HEALTH & PERSONAL
	HOME & GARDEN
	JEWELRY
	MUSIC
	MUSIC INSTRUMENTS
	OFFICE PRODUCTS
	SOFTWARE
	SPORTING GOODS
	TOOLS & HARDWARE
	TOYS
	VIDEO GAMES
	SHOPPING HOME

	MORE SHOPPING...

	CATEGORIES ABOUT P-ADIC NUMBER

	field theory

	number theory

Information About

P-adic Numbers