ℤ_{p}, ℚ_{p}, AND THE RING OF WITT VECTORS

YOSHIFUMI TSUCHIMOTO

No.02: definition of ℤ_{p} Let p be a prime (“base”). We would like to introduce a metric on

ℤ such that

n :small ⟺ n is divisible by powers of p.

Namely:

Definition 2.1. Let p be a prime number.

(1) We define a p-adic norm | • |_{p} on ℤ as follows.

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

(2) We define a p-adic distance d_{p}(•, •) on ℤ as follows.

d_{p}(n, m) = |n − m|_{p} (n, m ∈ ℤ)

Lemma 2.2. (1) | • |_{d} enjoys the following properties.

(a) |x|_{p} ≥ 0 (∀x ∈ ℤ). |x|_{p} = 0 ⟺ x = 0.

(b) |x + y|_{p} ≤ max(|x|_{p}, |y|_{p}) (≤ (|x|_{p} + |y|_{p}).

(c) |xy|_{p} = |x|_{p}|y|_{p}

(2) (ℤ, d_{p}) is a metric space.

Definition 2.3. A metric space (X, d) is said to be complete if every Cauchy sequence of X converges to an element of X.

Theorem 2.4. Let (X, d) be a metric space. There exists a complete metric space (X̄, d) with an isometry ι : X → X̄ such that X is dense in X̄. Furthermore, X̄ is unique up to a unique isometry.

Definition 2.5. Let (X, d) be a metric space. We call (X̄, d) as in the above theorem the completion of (X, d).

Definition 2.6. Let p be a prime number. We denote the completion of (ℤ, d_{p}) by (ℤ_{p}, d_{p}) and call it the ring of p-addic integers. Thus elements of ℤ_{p} are p-addic integers.

Theorem 2.7. ℤ_{p} has a unique structure of a topological ring. Namely,

(1) There exists unique continuous maps

+ : ℤ_{p} × ℤ_{p} → ℤ_{p}

(addition) and

× : ℤ_{p} × ℤ_{p} → ℤ_{p}

(multiplication) which are extensions of the usual addition and multiplication of ℤ.

(2) (ℤ_{p}, +, ×) is a commutative associative ring.

YOSHIFUMI TSUCHIMOTO

Definition 2.8. Let p be a prime number. For any sequence {a_{j}}_{j=0}^{∞} such that a_{j} ∈ {0, 1, 2, 3, …, p−1}, we consider a sequence {s_{n}} defined by

s_{n} = ∑_{j=0}^{n} a_{j}p^{j}.

Then the sequence {s_{n}} is a Cauchy sequence in ℤ_{p}. We denote the limit of the sequence as

[0.a_{0}a_{1}a_{2}a_{3} …]_{p}.

Exercise 2.1. compute

[0.1]_{3} + [0.2222 …]_{3}