The topology of the p-adic numbers 1

Let An=Z/pnZ, then An with the canonical epimorphism φn:AnAn1 defines a projective system. The projective limit Zp is the subring of
n1An
containing (xn) such that φn(xn)=xn1. Suppose we give An the discrete topology and An the product topology. By Tychonoff theorem, we have a compact space An.  We call Zp the p-adic integers.

Theorem 1 The p-adic integers Zp is closed subset of An.
Proof) We prove that the complement Zcp is open. Let (xn)Zcp, then we have some positive number m such that φ(xm)xm1. Consider the following set
U={x1}××{xm}×n>mAn
which is open because there are only finitely many proper subsets of An. Clearly xnU, and for any other (yn)U, we have that ym=xm,ym1=xm1φm(ym)ym1. This means that (ym)Zp. This shows that UZcp and hence Zp closed.

This turns Zp into a compact space when given the subspace topology.

We will, however, approach the topology of the p-adic integers by using a more number-theoretic method. By the fundamental theorem of arithmetic, for any integer m, there exists a unique n such that pn|m yet pn+1|m. We define the p-adic valuation vp:ZN by vp(m)=n. We let vp(0)=. The p-adic valuation satisfies the following properties,

Properties
     1) vp(xy)=vp(x)+vp(y)
     2) vp(x+y)min(vp(x),vp(y))

We have that x=pex,y=pfy where e=vp(x) and f=vp(y) with x,y not divisible by p. Then xy=pe+fxy with xy coprime to p. This show the first property. For the second property, we let e be the minimum with out loss of generosity. Then x+y=pe(x+pfey) but because p|xp|x+pfey this implies that vp(x+y)e=min(vp(x),vp(y)).

We extend vp to Q as follows. For all qQ,q=x/y for some integers x,y. vp(q)=vp(x)vp(y). In the case when x=0, we let n=0 for any integer nZ. It is easy to check well-definedness. Let x1/y1=x2/y2x1y2=y1x2.
vp(x1y2)=vp(y1x2)vp(x1)+vp(y2)=vp(y1)+vp(x2)vp(x1)vp(x2)=vp(y1)vp(y2)vp(x1/x2)=vp(y1/y2)
We define another map |x|p=pvp(x), then by the above properties, we have

Properties
     1) |x|p0 and equality if and only if x=0
     2) |xy|p=|x|p|y|p
     3) |x+y|pmax(|x|p,|y|p)|x|p+|y|p

pvp(xy)=pvp(x)vp(y)=pvp(x)pvp(y)
vp(x+y)min(vp(x),vp(y))vp(x+y)max(vp(x),vp(x))

with the have that px0 for all real number x give us the desired results. This makes μ(x,y)=|xy|p into a metric, hence Q becomes a metric space respect to μ. We may define Cauchy sequence via this absolute value and prove that the set of all Cauchy sequences is, in fact, a ring. The subset of all null sequences forms a maximal ideal, and the quotient ring becomes a field. This field is called a completion of Q via the valuation vp, and we denote it by Qp. We can prove that Qp can also be a metric space by extending vp further to Qp and contains Q as a dense subspace.

The set
Zp:={xQp | vp(x)0}
is a subring of Zp. The notation would not cause any confusion with our original p-adic integers Zp because we have that Zp defined using valuation and completion and Zp defined using projective limit are isomorphic (both algebraically and topologically).

Comments

Popular posts from this blog

Max-Spec vs Spec

함수와 무한대2