LemmaStatement: AI-adaptedProof: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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Uniqueness of an extended complete field absolute value

Statement

For a finite extension E/F of a complete absolutely valued field F, at most one absolute value on E extends the given absolute value on F. Any such extension makes E complete.

Facts & Assumptions

Given: The data and hypotheses of the statement.

[F1]

Finite dimensional norm equivalence over a complete valued field: Let F be complete for a multiplicative absolute value and V a finite-dimensional normed F-vector space. For any basis $v_{1}, \dots, v_{n}$ , its coordinate sup norm $∥ \sum a_{i} v_{i} ∥_{\infty} = \max_{i} ∣ a_{i} ∣$ is bounded above and below by positive multiples of the given norm. For $n = 0$ both norms are zero. Consequently V is complete and every linear subspace is closed.

Proof

1.1

Two extending absolute values are norms on the finite-dimensional F-vector space E. Norm equivalence gives constants $c, C > 0$ with $c ∣ x ∣_{1} \leq ∣ x ∣_{2} \leq C ∣ x ∣_{1}$ for every x. It also gives completeness for either norm.

2.1

For $x \neq 0$ , apply the comparison to $x^{n}$ and take nth roots: $c^{1 / n} ∣ x ∣_{1} \leq ∣ x ∣_{2} \leq C^{1 / n} ∣ x ∣_{1}$ . Let n tend to infinity to obtain equality. Both values of zero are zero. This includes the trivial valuation and E=F, and asserts uniqueness only if an extension exists.

step 1.1algebra∎

Depends on

Finite dimensional norm equivalence over a complete valued field

Used by

Number field places classification Theorem
Unique extension of a nonarchimedean absolute value Theorem

Dependency tree · two levels

2 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

§6, Lemma 6.1, pp.10–11 (standard reference, not scraped)