The topology of the p-adic numbers 1
Let \( A_n = \mathbb{Z}/p^n \mathbb{Z} \), then \( A_n \) with the canonical epimorphism \( \varphi_n: A_n \rightarrow A_{n-1} \) defines a projective system. The projective limit \( \mathbb{Z}_p \) is the subring of
\[
\prod_{n \geq 1} A_n
\]
containing \( (x_n) \) such that \( \varphi_n(x_n) = x_{n-1} \). Suppose we give \( A_n \) the discrete topology and \( \prod A_n \) the product topology. By Tychonoff theorem, we have a compact space \( \prod A_n \). We call \( \mathbb{Z}_p \) the \( p \)-adic integers.
Theorem 1 The \( p \)-adic integers \( \mathbb{Z}_p \) is closed subset of \( \prod A_n \).
Proof) We prove that the complement \( \mathbb{Z}_p^c \) is open. Let \( (x_n) \in \mathbb{Z}_p^c \), then we have some positive number \( m \) such that \( \varphi(x_m) \neq x_{m-1} \). Consider the following set
\[
U = \{ x_1 \} \times \cdots \times \{ x_m \} \times \prod_{n > m} A_n
\]
which is open because there are only finitely many proper subsets of \( A_n \). Clearly \( x_n \in U \), and for any other \( (y_n) \in U \), we have that \( y_m = x_m, y_{m-1} = x_{m-1} \Rightarrow \varphi_m(y_m) \neq y_{m-1} \). This means that \( (y_m) \notin \mathbb{Z}_p \). This shows that \(U \subset \mathbb{Z}_p^c \) and hence \( \mathbb{Z}_p \) closed.
This turns \( \mathbb{Z}_p \) 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 \( p^n | m \) yet \( p^{n+1} \not| m \). We define the \( p \)-adic valuation \( v_p: \mathbb{Z} \rightarrow \mathbb{N} \) by \( v_p(m) = n \). We let \( v_p(0) = \infty \). The \( p \)-adic valuation satisfies the following properties,
Properties
1) \( v_p(xy) = v_p(x) + v_p(y) \)
2) \( v_p(x+y) \geq \min(v_p(x), v_p(y)) \)
We have that \( x = p^ex', y= p^fy' \) where \( e= v_p(x) \) and \( f = v_p(y) \) with \( x', y' \) not divisible by \( p \). Then \( xy = p^{e+f}x'y' \) 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 \( x+y = p^e(x'+p^{f-e}y') \) but because \( p \not| x' \Rightarrow p \not| x'+ p^{f-e}y' \) this implies that \( v_p(x+y) \geq e = \min(v_p(x), v_p(y)) \).
We extend \( v_p \) to \( \mathbb{Q} \) as follows. For all \( q \in \mathbb{Q}, q = x/y \) for some integers \( x,y \). \( v_p(q) = v_p(x) - v_p(y) \). In the case when \( x = 0 \), we let \( \infty - n = 0 \) for any integer \( n \in \mathbb{Z} \). It is easy to check well-definedness. Let \( x_1/y_1 = x_2/y_2 \Rightarrow x_1y_2 = y_1x_2 \).
\[
\begin{array}{llcl}
& v_p(x_1y_2) & = & v_p(y_1x_2) \\
\Rightarrow & v_p(x_1) + v_p(y_2) & = & v_p(y_1) + v_p(x_2) \\
\Rightarrow & v_p(x_1) - v_p(x_2) & = & v_p(y_1) - v_p(y_2) \\
\Rightarrow & v_p(x_1/x_2) & = & v_p(y_1/y_2)
\end{array}
\]
We define another map \( |x|_p = p^{-v_p(x)} \), then by the above properties, we have
Properties
1) \( |x|_p \geq 0 \) and equality if and only if \( x = 0 \)
2) \( |xy|_p = |x|_p|y|_p \)
3) \( |x+y|_p \leq \max(|x|_p, |y|_p) \leq |x|_p + |y|_p \)
\(p^{-v_p(xy)} = p^{-v_p(x)-v_p(y)} = p^{-v_p(x)}p^{-v_p(y)} \)
\(v_p(x+y) \geq \min(v_p(x), v_p(y)) \Rightarrow -v_p(x+y) \leq \max(-v_p(x), -v_p(x)) \)
with the have that \( p^x \geq 0 \) for all real number \( x \) give us the desired results. This makes \( \mu(x,y) = |x-y|_p \) into a metric, hence \( \mathbb{Q} \) becomes a metric space respect to \( \mu \). 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 \( \mathbb{Q} \) via the valuation \( v_p \), and we denote it by \( \mathbb{Q}_p \). We can prove that \( \mathbb{Q}_p \) can also be a metric space by extending \( v_p \) further to \( \mathbb{Q}_p \) and contains \( \mathbb{Q} \) as a dense subspace.
The set
\[ \mathbb{Z}_p := \{ x \in \mathbb{Q}_p ~|~ v_p(x) \geq 0 \} \]
is a subring of \( \mathbb{Z}_p \). The notation would not cause any confusion with our original \( p \)-adic integers \( \mathbb{Z}_p \) because we have that \( \mathbb{Z}_p \) defined using valuation and completion and \( \mathbb{Z}_p \) defined using projective limit are isomorphic (both algebraically and topologically).
\[
\prod_{n \geq 1} A_n
\]
containing \( (x_n) \) such that \( \varphi_n(x_n) = x_{n-1} \). Suppose we give \( A_n \) the discrete topology and \( \prod A_n \) the product topology. By Tychonoff theorem, we have a compact space \( \prod A_n \). We call \( \mathbb{Z}_p \) the \( p \)-adic integers.
Theorem 1 The \( p \)-adic integers \( \mathbb{Z}_p \) is closed subset of \( \prod A_n \).
Proof) We prove that the complement \( \mathbb{Z}_p^c \) is open. Let \( (x_n) \in \mathbb{Z}_p^c \), then we have some positive number \( m \) such that \( \varphi(x_m) \neq x_{m-1} \). Consider the following set
\[
U = \{ x_1 \} \times \cdots \times \{ x_m \} \times \prod_{n > m} A_n
\]
which is open because there are only finitely many proper subsets of \( A_n \). Clearly \( x_n \in U \), and for any other \( (y_n) \in U \), we have that \( y_m = x_m, y_{m-1} = x_{m-1} \Rightarrow \varphi_m(y_m) \neq y_{m-1} \). This means that \( (y_m) \notin \mathbb{Z}_p \). This shows that \(U \subset \mathbb{Z}_p^c \) and hence \( \mathbb{Z}_p \) closed.
This turns \( \mathbb{Z}_p \) 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 \( p^n | m \) yet \( p^{n+1} \not| m \). We define the \( p \)-adic valuation \( v_p: \mathbb{Z} \rightarrow \mathbb{N} \) by \( v_p(m) = n \). We let \( v_p(0) = \infty \). The \( p \)-adic valuation satisfies the following properties,
Properties
1) \( v_p(xy) = v_p(x) + v_p(y) \)
2) \( v_p(x+y) \geq \min(v_p(x), v_p(y)) \)
We have that \( x = p^ex', y= p^fy' \) where \( e= v_p(x) \) and \( f = v_p(y) \) with \( x', y' \) not divisible by \( p \). Then \( xy = p^{e+f}x'y' \) 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 \( x+y = p^e(x'+p^{f-e}y') \) but because \( p \not| x' \Rightarrow p \not| x'+ p^{f-e}y' \) this implies that \( v_p(x+y) \geq e = \min(v_p(x), v_p(y)) \).
We extend \( v_p \) to \( \mathbb{Q} \) as follows. For all \( q \in \mathbb{Q}, q = x/y \) for some integers \( x,y \). \( v_p(q) = v_p(x) - v_p(y) \). In the case when \( x = 0 \), we let \( \infty - n = 0 \) for any integer \( n \in \mathbb{Z} \). It is easy to check well-definedness. Let \( x_1/y_1 = x_2/y_2 \Rightarrow x_1y_2 = y_1x_2 \).
\[
\begin{array}{llcl}
& v_p(x_1y_2) & = & v_p(y_1x_2) \\
\Rightarrow & v_p(x_1) + v_p(y_2) & = & v_p(y_1) + v_p(x_2) \\
\Rightarrow & v_p(x_1) - v_p(x_2) & = & v_p(y_1) - v_p(y_2) \\
\Rightarrow & v_p(x_1/x_2) & = & v_p(y_1/y_2)
\end{array}
\]
We define another map \( |x|_p = p^{-v_p(x)} \), then by the above properties, we have
Properties
1) \( |x|_p \geq 0 \) and equality if and only if \( x = 0 \)
2) \( |xy|_p = |x|_p|y|_p \)
3) \( |x+y|_p \leq \max(|x|_p, |y|_p) \leq |x|_p + |y|_p \)
\(p^{-v_p(xy)} = p^{-v_p(x)-v_p(y)} = p^{-v_p(x)}p^{-v_p(y)} \)
\(v_p(x+y) \geq \min(v_p(x), v_p(y)) \Rightarrow -v_p(x+y) \leq \max(-v_p(x), -v_p(x)) \)
with the have that \( p^x \geq 0 \) for all real number \( x \) give us the desired results. This makes \( \mu(x,y) = |x-y|_p \) into a metric, hence \( \mathbb{Q} \) becomes a metric space respect to \( \mu \). 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 \( \mathbb{Q} \) via the valuation \( v_p \), and we denote it by \( \mathbb{Q}_p \). We can prove that \( \mathbb{Q}_p \) can also be a metric space by extending \( v_p \) further to \( \mathbb{Q}_p \) and contains \( \mathbb{Q} \) as a dense subspace.
The set
\[ \mathbb{Z}_p := \{ x \in \mathbb{Q}_p ~|~ v_p(x) \geq 0 \} \]
is a subring of \( \mathbb{Z}_p \). The notation would not cause any confusion with our original \( p \)-adic integers \( \mathbb{Z}_p \) because we have that \( \mathbb{Z}_p \) defined using valuation and completion and \( \mathbb{Z}_p \) defined using projective limit are isomorphic (both algebraically and topologically).
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