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p-adic Arithmetic
Stany De Smedt

Introduction

Let be a prime number, fixed once and for all. If is any rational number other than , we can write in the form , where are relatively prime to and . We now define and , and and . They satisfy the following properties.

1. if and only if .

2. (the strong triangle inequality) with if (the isosceles triangle principle).

3. .

With instead of , we get for 2 and 3:

is called the p-adic valuation; is called the p-adic order.

Ostrowski proved that each nontrivial valuation on the field of rational numbers is equivalent either to the absolute value function or to some p-adic valuation.

Recall that two valuations and are equivalent if there exists a positive constant such that . For a proof of this theorem, see [1].

The completion of the field of rational numbers with respect to the p-adic valuation is called the field of p-adic numbers and will be denoted . The set is the ring of p-adic integers.

It can be easily proved that each p-adic number can be written in the form

where each is one of the elements , and . This is called the Hensel representation of the p-adic numbers. Sometimes one uses, analogous to the ordinary decimal notation for real numbers, the notation

It is this representation, preceded with to denote the p-adic, that we will use in our package.

With this representation, one obtains for :

and

The set of values of is , and is called the valuation group of .

Every can be written as with . The elements with for sufficiently large can be identified with the nonnegative integers. Thus and also contains as a subset. For example,

Note also that every nonzero element of can be written as , where and with . And just as with decimal fractions, a p-adic number is rational if and only if the sequence of the digits is periodic from some index on.

Due to the strong triangle inequality, we may use the following property, which is much easier than in the Archimedean case:

In general we say that a sequence in converges to if and only if and we will write this as .

For and we have the following topological properties, which we only present here for completeness.

1. is compact: each covering by means of open sets has a finite subcovering.

2. is complete: every Cauchy sequence converges.

3. is dense in : each element of is the limit of elements of .

4. is locally compact: each point has a compact neighborhood.

5. is dense in .

6. is complete and separable: it has a countable dense subset.

7. is zero-dimensional: every neighborhood of a point contains an open and closed subset.

8. is totally disconnected: the only subsets that are connected as a metric space are the empty set and the singletons.

We now survey the different parts that are treated in this package. For more information on p-adic calculus, see [1] or [2].



     
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