p-adic number
Every rational number can be uniquely expressed as the sum of a series as above, with respect to the p-adic absolute value.
[note 2] Roughly speaking, modular arithmetic modulo a positive integer n consists of "approximating" every integer by the remainder of its division by n, called its residue modulo n. The main property of modular arithmetic is that the residue modulo n of the result of a succession of operations on integers is the same as the result of the same succession of operations on residues modulo n. If one knows that the absolute value of the result is less than n/2, this allows a computation of the result which does not involve any integer larger than n. For larger results, an old method, still in common use, consists of using several small moduli that are pairwise coprime, and applying the Chinese remainder theorem for recovering the result modulo the product of the moduli.
If the process is continued infinitely, this provides eventually a result which is a p-adic number.
The proof of the lemma results directly from the fundamental theorem of arithmetic.
Every rational number may be viewed as a p-adic series with a single nonzero term, consisting of its factorization of the form
Iterating this process, possibly infinitely many times, provides eventually the desired normalized p-adic series.
The one that is given here is relatively elementary, since it does not involve any other mathematical concepts than those introduced in the preceding sections.
Other equivalent definitions use completion of a discrete valuation ring (see § p-adic integers), completion of a metric space (see § Topological properties), or inverse limits (see § Modular properties).
However, the general construction of a completion can be simplified in this case, because the metric is defined by a discrete valuation (in short, one can extract from every Cauchy sequence a subsequence such that the differences between two consecutive terms have strictly decreasing absolute values; such a subsequence is the sequence of the partial sums of a p-adic series, and thus a unique normalized p-adic series can be associated to every equivalence class of Cauchy sequences; so, for building the completion, it suffices to consider normalized p-adic series instead of equivalence classes of Cauchy sequences).
As the metric is defined from a discrete valuation, every open ball is also closed.
This expansion can be computed by long division of the numerator by the denominator, which is itself based on the following theorem: If
The decimal expansion is obtained by repeatedly applying this result to the remainder
The p-adic expansion of a rational number is defined similarly, but with a different division step.
is the formal power series obtained by repeating indefinitely the above division step on successive remainders.
, the process stops eventually with a zero remainder; in this case, the series is completed by trailing terms with a zero coefficient, and is the representation of
The existence and the computation of the p-adic expansion of a rational number results from Bézout's identity in the following way.
This means that the production of the digits is reversed and the limit happens on the left hand side.
(the factor 5 has to be viewed as a "shift" of the p-adic valuation, similar to the basis of any number expansion, and thus it should not be itself expanded).
It is possible to use a positional notation similar to that which is used to represent numbers in base p. Let
consecutively, ordered by decreasing values of i, often with p appearing on the right as an index: So, the computation of the example above shows that and When
So far this article has used a notation for p-adic expansions in which powers of p increase from right to left.
for example, is written as When performing arithmetic in this notation, digits are carried to the left.
It is also possible to write p-adic expansions so that the powers of p increase from left to right, and digits are carried to the right.
Also contrasting the case of real numbers, although there is a unique extension of the p-adic valuation to
Given a natural number k, the index of the multiplicative group of the k-th powers of the non-zero elements of
The number e, defined as the sum of reciprocals of factorials, is not a member of any p-adic field; but
Pick a non-zero prime ideal P of D. If x is a non-zero element of E, then xD is a fractional ideal and can be uniquely factored as a product of positive and negative powers of non-zero prime ideals of D. We write ordP(x) for the exponent of P in this factorization, and for any choice of number c greater than 1 we can set Completing with respect to this absolute value |⋅|P yields a field EP, the proper generalization of the field of p-adic numbers to this setting.
For example, when E is a number field, Ostrowski's theorem says that every non-trivial non-Archimedean absolute value on E arises as some |⋅|P.
The remaining non-trivial absolute values on E arise from the different embeddings of E into the real or complex numbers.
(In fact, the non-Archimedean absolute values can be considered as simply the different embeddings of E into the fields Cp, thus putting the description of all the non-trivial absolute values of a number field on a common footing.)
