ℤ_{p}, ℚ_{p}, AND THE RING OF WITT VECTORS No.04: Review There are several ways to define ℤ_{p}:

(1) ℤ_{p} is the completion of ℤ with respect to the p-adic metric d_{p}.

(2) ℤ_{p} is the projective limit \underleftarrow{lim}_{k}(ℤ/p^{k}ℤ). (3)

ℤ_{p} = {[0.a_{0}a_{1}a_{2}...]_{p}; a_{i} ∈ {0, 1, 2, 3, …, p − 1}}

Definition 4.1. We equip

\underleftarrow{lim}_{k}(ℤ/p^{k}ℤ) = \{(b_{j})_{j=1}^{∞}; b_{j} ∈ ℤ/p^{j}ℤ, (b_{j_{1}} modulo p^{j_{2}}) = b_{j_{2}} whenever j_{1} > j_{2}\}

with the following “p-addic norm”.

|(b_{j})|_{p} = \begin{cases}\frac{1}{p^{k}} & if b_{k} = 0 and b_{k+1} ≠ 0 \\ 0 & if (b_{j}) = 0 \end{cases}

Then we define “p-addic metric” d_{p} on \underleftarrow{lim}_{k} ℤ/p^{k}ℤ in an obvious way.

Lemma 4.2. A natural map ι : (ℤ, d_{p}) → (\underleftarrow{lim}_{k}(ℤ/p^{k}ℤ), d_{p}) defined by

ι : ℤ ∋ n ↦ (n, n, n …) ∈ \underleftarrow{lim}_{k}(ℤ/p^{k}ℤ)

is an isometry.

Exercise 4.1. Prove the lemma above.