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 Qp 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 Zp be defined by taking as a basis all sets of the form {n + λ pa for λ in Zp and a in N}. Then Zp 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 Zp 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. Qp 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 Qp, it is not (metrically) complete. Its (metric) completion is called Cp or Ωp. Here an end is reached, as Cp is algebraically closed. Unlike the complex field, Cp is not locally compact.

The field Cp is algebraically isomorphic to the field C of complex numbers, so we may regard Cp 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 Qp in the multiplicative group of Qp 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 Qp. For instance, the function

f: QpQp, 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 rp is in Qp (and Q stands for R), it is possible to find a sequence (xn) in Q such that for all p (including ∞), the limit of xn in Qp is rp.

The field Qp is a locally compact Hausdorff space.

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

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