Global Fields - Local Fields - Cambridge III notes

Definition 8.1 (Global field). A global field is a field which is either:

(i) An algebraic number field
(ii) A global function field, i.e. a finite extension of $𝔽_{p} (t)$ .

Lemma 8.2. Assuming that:

$(K, | ∙ |)$ is a complete discretely valued field
$L ∕ K$ a finite Galois extension with absolute value ${| ∙ |}_{L}$ extending $| ∙ |$ .

Then for

x \in L

and

σ \in Gal (L ∕ K)

, we have

{| σ (x) |}_{L} = {| x |}_{L}

Proof. Since $x \mapsto {| σ (x) |}_{L}$ is another absolute value on $L$ extending $| ∙ |$ on $K$ , the result follows from uniqueness of ${| ∙ |}_{L}$ . □

Lemma 8.3 (Kummer’s Lemma). Assuming that:

$(K, | ∙ |)$ a complete discretely valued field
$f (X) \in K [X]$ a separable irreducible polynomial with roots $α_{1}, \dots, α_{n} \in K^{sep}$ ( $K^{sep}$ is the separable closure of $K$ )
$β \in K^{sep}$ with
$| β - α_{1} | < | β - α_{i} |$

for $i = 2, \dots, n$ .

Then

α_{1} \in K (β)

Proof. Let $L = K (β)$ , $L^{'} = L (α_{1}, \dots, α_{n})$ . Then $L^{'} ∕ L$ is a Galois extension. Let $σ \in Gal (L^{'} ∕ L)$ . We have

\begin{array}{l} | β - σ (α_{1}) | & = | σ (β - α_{1}) | \\ = | β - α_{1} | \end{array}

using Lemma 8.2. Hence $σ (α_{1}) = α_{1}$ , so $α_{1} \in K (β)$ . □

Proposition 8.4. Assuming that:

$(F, | ∙ |)$ is a complete discretely valued field
$f (X) = \sum_{i = 0}^{n} a_{i} X^{i} \in O_{K} [X]$ a separable irreducible monic polynomial
$α \in K^{sep}$ a root of $f$

Then there exists

𝜀 > 0

such that for any

g (X) = \sum_{i = 0}^{n} b_{i} X^{i} \in O_{K} [X]

monic with

| a_{i} - b_{i} | < 𝜀

for all

i

, there exists a root

β

g (X)

such that

K (α) = K (β)

Proof. Let $α_{1}, \dots, α_{n} \in K^{sep}$ be the roots of $f$ which are necessarily distinct. Then $f^{'} (α_{1}) \neq 0$ . We choose $𝜀$ sufficiently small such that $| g (α_{1}) | < | f^{'} (α_{1}) |^{2}$ and $| f^{'} (α_{1}) - g^{'} (α_{1}) | < | f^{'} (α_{1}) |$ . Then we have $| g^{'} (α_{1}) | < | f^{'} (α_{1}) |^{2} = | g^{'} (α_{1}) |^{2}$ (the equality is by Lemma 1.6).

By Hensel’s Lemma version 1 applied to the field $K (α_{1})$ there exists $β \in K (α_{1})$ such that $g (β) = 0$ and $| ∙ - α_{1} | < | g^{'} (α_{1}) |$ . Then

\begin{array}{l} | g^{'} (α_{1}) | & = | f^{'} (α_{1}) | \\ = \prod_{j = 1}^{n} | α_{1} - α_{j} | \\ \leq | α_{1} - α_{i} | \end{array}

for $i = 2, \dots, n$ . (Use $| α_{1} - α_{i} | \leq 1$ since $α_{i}$ integral). Since $| β - α_{1} | < | α_{1} - α_{i} | = | β - α_{i} |$ using Lemma 1.6, we have that Kummer’s Lemma gives that $α_{1} \in K (β)$ and hence $K (α_{1}) = K (β)$ . □

Theorem 8.5. Assuming that:

$K$ is a local field

Then

K

is the completion of a global field.

Proof. Case 1: $| ∙ |$ is archimedean. Then $ℝ$ is the completion of $ℚ$ , and $ℂ$ is the completion of $ℚ (i)$ (with respect to ${| ∙ |}_{\infty}$ ).

Case 2: $| ∙ |$ non-archimedean, equal characteristic. Then $K ≅ 𝔽_{q} ((t))$ is the completion of $𝔽_{q} (t)$ with respect to the $t$ -adic valuation.

Case 3: $| ∙ |$ non-archimedean mixed characteristic. Then $K = ℚ_{p} (α)$ , with $α$ a root of a monic irreducible polynomial $f (X) \in ℤ_{p} [X]$ . Since $ℤ$ is dense in $ℤ_{p}$ , we choose $g (X) \in ℤ [X]$ as in Proposition 8.4. Then $K = ℚ (β)$ with $β$ a root of $g (X)$ . Since $ℚ (β)$ dense in $ℚ_{p} (β) = K$ , and $K$ is complete, we must have that $K$ is the completion of $ℚ (β)$ . □

8 Global Fields