ℤ_{p}, ℚ_{p}, AND WITT ALGEBRAS

No.06: ℚ_{p}

Definition 6.1. We denote by ℚ_{p} the quotient field of ℤ_{p}.

Lemma 6.2. Every non zero element x ∈ ℚ_{p} is uniquely expressed as

x = p^{k}u (k ∈ ℤ, u ∈ ℚ_{p}^{×}).

We have so far constructed a ring ℤ_{p} and a field ℚ_{p} for each prime p.

Proposition 6.3. Let p be a prime. Then:

(1) ℤ_{p} is a local ring with the unique maximal ideal pℤ_{p}. (2)

ℤ_{p}/pℤ_{p} ≅ 𝔽_{p}(= ℤ/pℤ).

(3) ℤ_{p} is an integral domain whose quotient field ℚ_{p} is a field of \underline{characteristic zero}.

With ℚ_{p} and/or ℤ_{p}, we may do some “calculus” such as:

Theorem 6.4. [1, corollary 1 of theorem 1] Let f ∈ ℤ_{p}[X_{1}, X_{2}, …, X_{m}], x ∈ ℤ_{p}^{m} , n, k ∈ ℤ. Assume that there exists a natural number j such that 1 ≤ j ≤ m,

\frac{∂f}{∂X_{j}} (x) ≢ 0 (mod p).

Then there exists y ∈ ℤ_{p}^{m} such that

f(y) = 0 (1) y ≡ x (mod p) (2)

See [1] for details.

References

[1] J. P. Serre, Cours d’arithmétique, Presses Universitaires de France, 1970.