PropositionStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

A finite extension has degree one if and only if the two fields are equal

Statement

For a finite field extension $K / F$ ,

$[K : F] = 1 ⟺ K = F .$

Facts & Assumptions

Given: A finite extension $K / F$ .

[L1]

The degree $[K : F]$ is the dimension of $K$ as an $F$ -vector space (The degree $[K : F] = \dim_{F} K$ of a finite field extension).

[L2]

With respect to a one-element basis, every vector has a unique one-coordinate expression (A finite list $v : n \to V$ is an ordered basis if and only if every $x \in V$ equals $\sum_{i < n} λ_{i} v_{i}$ for exactly one $λ : n \to F$ ; those scalars are the coordinates of $x$ in that ordered basis).

Proof

technique · direct

1.1givenL1L2choose

Suppose $[K : F] = 1$ and choose a one-element basis $(b)$ . By [L2], write $1_{K} = c b$ with $c \in F$ . Since $1_{K} \neq 0$ , one has $c \neq 0$ , so $b = c^{- 1} 1_{K}$ . Every $x \in K$ is therefore of the form $d b = (d c^{- 1}) 1_{K}$ with $d \in F$ , and hence lies in the embedded copy of $F$ .

1.2L1

Conversely, if $K = F$ , then $(1_{F})$ is a basis of $F$ over itself, so [L1] gives $[F : F] = 1$ .

2.1step 1.1given

Step 1.1 gives $K \subseteq F$ , while the extension already has $F \subseteq K$ , so $K = F$ .

3.1step 2.1step 1.2∎

Steps 1.1, 1.2, and 2.1 prove both directions.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

19 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

A. W. Knapp, Basic Algebra, 2nd ed., Chapter IX, Section 1 (standard reference, not scraped)