p-adic Number - p-adic Expansions

p-adic Expansions

When dealing with ordinary real numbers, if we take p to be a fixed prime number, then any positive integer can be written as a base p expansion in the form

where the a_i are integers in {0, … , p − 1}. For example, the 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 extending this description to the larger domain of the rationals (and, ultimately, to the reals) is to use sums of the form:

A definite meaning is given to these sums based on Cauchy sequences, 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 for which a_i = 0 for all i < 0.

With p-adic numbers, on the other hand, we choose to extend the base p expansions in a different way. Because in the p-adic world high positive powers of p are small and high negative powers are large, we consider infinite sums of the form:

where k is some (not necessarily positive) integer. With this approach we obtain the p-adic expansions of the p-adic numbers. Those p-adic numbers for which a_i = 0 for all i < 0 are also called the p-adic integers.

As opposed to real number expansions which extend to the right as sums of ever smaller, increasingly negative powers of the base p, p-adic numbers may expand to the left forever, a property that can often be true for the p-adic integers. For example, consider the p-adic expansion of 1/3 in base 5. It can be shown to be …1313132₅, i.e., the limit of the sequence 2₅, 32₅, 132₅, 3132₅, 13132₅, 313132₅, 1313132₅, … :

Multiplying this infinite sum by 3 in base 5 gives …0000001₅. 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 satisfies the definition of being a p-adic integer in base 5.

More formally, the p-adic expansions can be used to define the field Q_p of p-adic numbers while the p-adic integers form a subring of Q_p, denoted Z_p. (Not to be confused with the ring of integers modulo p which is also sometimes written Z_p. To avoid ambiguity, Z/pZ or Z/(p) are often used to represent the integers modulo p.)

While it is possible to use the approach above to define p-adic numbers and explore their properties, just as in the case of real numbers other approaches are generally preferred. Hence we want to define a notion of infinite sum which makes these expressions meaningful, and this is most easily accomplished by the introduction of the p-adic metric. Two different but equivalent solutions to this problem are presented in the Constructions section below.