p-adic Number - Properties

Properties

The ring of p-adic integers is the inverse limit of the finite rings Z/pkZ, but is nonetheless uncountable, and has the cardinality of the continuum. Accordingly, the field Q_p is uncountable. The endomorphism ring of the Prüfer p-group of rank n, denoted Z(p∞)n, is the ring of n×n matrices over the p-adic integers; this is sometimes referred to as the Tate module.

The p-adic numbers contain the rational numbers Q and form a field of characteristic 0. This field cannot be turned into an ordered field.

Let the topology τ on Z_p be defined by taking as a basis all sets of the form {n + λ pa for λ in Z_p and a in N}. Then Z_p is a compactification of Z, under the derived topology (it is not a compactification of Z with its usual discrete topology). The relative topology on Z as a subset of Z_p is called the p-adic topology on Z.

The topology of the set of p-adic integers is that of a Cantor set; the topology of the set of p-adic numbers is that of a Cantor set minus a point (which would naturally be called infinity). In particular, the space of p-adic integers is compact while the space of p-adic numbers is not; it is only locally compact. As metric spaces, both the p-adic integers and the p-adic numbers are complete.

The real numbers have only a single proper algebraic extension, the complex numbers; in other words, this quadratic extension is already algebraically closed. By contrast, the algebraic closure of the p-adic numbers has infinite degree, i.e. Q_p has infinitely many inequivalent algebraic extensions. Also contrasting the case of real numbers, although there is a unique extension of the p-adic valuation to the algebraic closure of Q_p, it is not (metrically) complete. Its (metric) completion is called C_p or Ω_p. Here an end is reached, as C_p is algebraically closed. Unlike the complex field, C_p is not locally compact.

The field C_p is algebraically isomorphic to the field C of complex numbers, so we may regard C_p as the complex numbers endowed with an exotic metric. It should be noted that the proof of existence of such a field isomorphism relies on the axiom of choice, and does not provide an explicit example of such an isomorphism.

The p-adic numbers contain the nth cyclotomic field (n > 2) if and only if n divides p − 1. For instance, the nth cyclotomic field is a subfield of Q₁₃ if and only if n = 1, 2, 3, 4, 6, or 12. In particular, there is no multiplicative p-torsion in the p-adic numbers, if p > 2. Also, −1 is the only non-trivial torsion element in 2-adic numbers.

Given a natural number k, the index of the multiplicative group of the kth powers of the non-zero elements of Q_p in the multiplicative group of Q_p is finite.

The number e, defined as the sum of reciprocals of factorials, is not a member of any p-adic field; but ep is a p-adic number for all p except 2, for which one must take at least the fourth power. (Thus a number with similar properties as e – namely a pth root of ep – is a member of the algebraic closure of the p-adic numbers for all p.)

For reals, the only functions whose derivative is zero are the constant functions. This is not true over Q_p. For instance, the function

f: Q_p → Q_p, f(x) = (1/|x|_p)2 for x ≠ 0, f(0) = 0,

has zero derivative everywhere but is not even locally constant at 0.

Given any elements r_∞, r₂, r₃, r₅, r₇, ... where r_p is in Q_p (and Q_∞ stands for R), it is possible to find a sequence (x_n) in Q such that for all p (including ∞), the limit of x_n in Q_p is r_p.

The field Q_p is a locally compact Hausdorff space.

If is a finite Galois extension of, the Galois group is solvable. Thus, the Galois group is prosolvable.