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Hilbert's theorem 90 for a finite cyclic extension

Statement

Let $K / F$ be a finite cyclic extension of degree $n$ with $Gal (K / F) = ⟨ σ ⟩$ . For $b \in K^{\times}$ , the following are equivalent:

$N_{K / F} (b) = 1$ .
There exists $β \in K^{\times}$ with $b = \frac{β}{σ (β)} .$

Facts & Assumptions

Given: A finite cyclic extension $K / F$ of degree $n$ , a generator $σ$ of its Galois group, and an element $b \in K^{\times}$ .

[F1]

A cyclic extension is a finite Galois extension with cyclic Galois group (A cyclic extension is a finite Galois extension with cyclic Galois group).

[L1]

In a finite Galois extension, the norm is the product over the distinct $F$ -embeddings (Norm and trace from embeddings, with the inseparable exponent in the norm formula).

[L2]

Distinct characters of a group into a field are linearly independent (Dedekind's linear independence theorem for distinct characters).

Proof

technique · direct

1.1F1L1

For the forward direction from 2 to 1, suppose $b = β / σ (β)$ for some $β \in K^{\times}$ . Since the embeddings of $K / F$ are $1, σ, \dots, σ^{n - 1}$ , [L1] gives $N_{K / F} (b) = \prod_{i = 0}^{n - 1} \frac{σ^{i} (β)}{σ^{i + 1} (β)} = 1,$ the numerator and denominator cancelling cyclically because $σ^{n} = 1$ .

1.2F1L2choose

For the converse, assume $N_{K / F} (b) = 1$ . For $0 \leq i < n$ , set $c_{i} : = \prod_{j = 0}^{i - 1} σ^{j} (b),$ with the empty product $c_{0} = 1$ . Then the distinct automorphisms $1, σ, \dots, σ^{n - 1}$ restrict to distinct characters $K^{\times} \to K^{\times}$ , so [L2] implies that the $K$ -linear operator $T (x) : = \sum_{i = 0}^{n - 1} c_{i} σ^{i} (x)$ is not identically zero. Choose $x \in K$ with $β : = T (x) \neq 0$ .

2.1step 1.2algebra

The coefficients satisfy $c_{i + 1} = c_{i} σ^{i} (b)$ for $0 \leq i < n - 1$ , and $c_{n} = N_{K / F} (b) = 1$ . Therefore $σ (c_{i}) = b^{- 1} c_{i + 1} (0 \leq i < n - 1), σ (c_{n - 1}) = b^{- 1} .$ Applying $σ$ to $β = T (x)$ and re-indexing the sum gives $σ (β) = b^{- 1} β .$ Hence $b = β / σ (β)$ .

3.1step 1.1step 2.1∎

Steps 1.1 and 2.1 prove the equivalence.

Remarks

The proof uses only Dedekind independence. No cohomological language is needed here, although this is the classical vanishing of $H^{1}$ for a finite cyclic extension.

Depends on

Used by

Hilbert 90 for ℚ(i)/ℚ recovers the rational parametrization of the unit circle Example

Dependency tree · two levels

17 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

J. S. Milne, Fields and Galois Theory, v5.10, Theorem 5.23 and Corollary 5.25 (standard reference, not scraped)
S. R. Ghorpade, Lectures on Field Theory and Ramification Theory, Section 1.3 (standard reference, not scraped)