P-adic number

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.

Constructions

Analytic approach

The real numbers can be defined as equivalence classes of Cauchy sequences of rational numbers; 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.

Related Topics:
Real number - Equivalence class - Cauchy sequence - Rational number - Metric - Euclidean metric

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

for any non-zero rational number x, there is a unique integer n allowing us to write x = pn(a/b), where neither of the integers a and b is divisible by p. Unless the numerator or denominator of x contains a factor of p, n will be 0. Now define |x|p = p−n. We also define |0|p = 0.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

For example with x = 63/550 = 2−1 32 5−2 7 11−1

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:|x|_2=2

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:|x|_3=1/9

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:|x|_5=25

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:|x|_7=1/7

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:|x|_{11}=11

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:|x|_{mbox{any other prime}}=1

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

This definition of |x|p has the effect that high powers of p become "small".

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

It can be proved that each norm on Q is equivalent either to the Euclidean norm or to one of the p-adic norms for some prime p. The p-adic norm defines a metric dp on Q by setting

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

:d_p(x,y)=|x-y|_p

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The field Qp of p-adic numbers can then be defined as the completion of the metric space (Q,dp); its elements are equivalence classes of Cauchy sequences, where two sequences are called equivalent if their difference converges to zero. In this way, we obtain a complete metric space which is also a field and contains Q.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

It can be shown that in Qp, every element x may be written in a unique way as

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

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

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

where k is some integer and each ai is in {0,...,p − 1}. This series converges to x with respect to the metric dp.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Algebraic approach

In the algebraic approach, we first define the ring of p-adic integers, and then construct the field of quotients of this ring to get the field of p-adic numbers.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

We start with the inverse limit of the rings

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Z/pnZ (see modular arithmetic): a p-adic integer is then a sequence

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

(an)n≥1 such that an is in

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Z/pnZ, and if n < m,

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

an ? am (mod pn).

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Every natural number m defines such a sequence (m mod pn), and can therefore be regarded as a p-adic integer. For example, in this case 35 as a 2-adic integer would be written as the sequence {1, 3, 3, 3, 3, 35, 35, 35, ...}.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Note that pointwise addition and multiplication of such sequences is well defined, since addition and multiplication commute with the mod operator, see modular arithmetic. Also, every sequence (an) where the first element is not 0 has an inverse: since in that case, for every n, an and p are coprime, and so an and pn are relatively prime. Therefore, each an has an inverse mod pn, and the sequence of these inverses, (bn), is the sought inverse of (an).

Related Topics:
Modular arithmetic - Coprime

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

Every such sequence can alternatively be written as a series of the form we considered above. For instance, in the 3-adics, the sequence (2, 8, 8, 35, 35, ...) can be written as 2 + 2*3 + 0*32 + 1*33 + 0*34 + ... The partial sums of this latter series are the elements of the given sequence.

~ ~ ~ ~ ~ ~ ~ ~ ~ ~

The ring of p-adic integers has no zero divisors, so we can take the quotient field to get the field Qp of p-adic numbers. Note that in this quotient field, every number can be uniquely written as p−nu with a natural number n and a p-adic integer u.

Related Topics:
Quotient field - Natural number

~ ~ ~ ~ ~ ~ ~ ~ ~ ~