The topology of the p-adic numbers 1
Let An=Z/pnZ, then An with the canonical epimorphism φn:An→An−1 defines a projective system. The projective limit Zp is the subring of
∏n≥1An
containing (xn) such that φn(xn)=xn−1. 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)≠xm−1. Consider the following set
U={x1}×⋯×{xm}×∏n>mAn
which is open because there are only finitely many proper subsets of An. Clearly xn∈U, and for any other (yn)∈U, we have that ym=xm,ym−1=xm−1⇒φm(ym)≠ym−1. This means that (ym)∉Zp. This shows that U⊂Zcp 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:Z→N 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+fx′y′ with x′y′ 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′+pf−ey′) but because p⧸|x′⇒p⧸|x′+pf−ey′ this implies that vp(x+y)≥e=min(vp(x),vp(y)).
We extend vp to Q as follows. For all q∈Q,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 n∈Z. It is easy to check well-definedness. Let x1/y1=x2/y2⇒x1y2=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=p−vp(x), then by the above properties, we have
Properties
1) |x|p≥0 and equality if and only if x=0
2) |xy|p=|x|p|y|p
3) |x+y|p≤max(|x|p,|y|p)≤|x|p+|y|p
p−vp(xy)=p−vp(x)−vp(y)=p−vp(x)p−vp(y)
vp(x+y)≥min(vp(x),vp(y))⇒−vp(x+y)≤max(−vp(x),−vp(x))
with the have that px≥0 for all real number x give us the desired results. This makes μ(x,y)=|x−y|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:={x∈Qp | 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).
∏n≥1An
containing (xn) such that φn(xn)=xn−1. 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)≠xm−1. Consider the following set
U={x1}×⋯×{xm}×∏n>mAn
which is open because there are only finitely many proper subsets of An. Clearly xn∈U, and for any other (yn)∈U, we have that ym=xm,ym−1=xm−1⇒φm(ym)≠ym−1. This means that (ym)∉Zp. This shows that U⊂Zcp 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:Z→N 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+fx′y′ with x′y′ 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′+pf−ey′) but because p⧸|x′⇒p⧸|x′+pf−ey′ this implies that vp(x+y)≥e=min(vp(x),vp(y)).
We extend vp to Q as follows. For all q∈Q,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 n∈Z. It is easy to check well-definedness. Let x1/y1=x2/y2⇒x1y2=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=p−vp(x), then by the above properties, we have
Properties
1) |x|p≥0 and equality if and only if x=0
2) |xy|p=|x|p|y|p
3) |x+y|p≤max(|x|p,|y|p)≤|x|p+|y|p
p−vp(xy)=p−vp(x)−vp(y)=p−vp(x)p−vp(y)
vp(x+y)≥min(vp(x),vp(y))⇒−vp(x+y)≤max(−vp(x),−vp(x))
with the have that px≥0 for all real number x give us the desired results. This makes μ(x,y)=|x−y|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:={x∈Qp | 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).
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