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TheoremStatement: Literature-sourcedProof: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17
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  • 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.
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The elements separable over the base form an intermediate field

Statement

For an algebraic extension K/F, the set

Ks:={a∈K:a is separable over F}

is an intermediate field between F and K.

Facts & Assumptions

Given: An algebraic extension K/F and separable elements a,b∈K.

[L1]

An algebraic extension generated by separable elements is separable (An algebraic extension generated by separable elements is separable).

[L2]

The subfield criterion requires 0,1, closure under subtraction and multiplication, and inverses of nonzero elements (Subfield: a subring of a field closed under inverses of its nonzero elements, and therefore a field with the restricted operations).

Proof

technique · direct
1.1

The extension F(a,b)/F is generated by separable elements, so [L1] makes every element of F(a,b) separable over F.

L1
2.1

In particular, a−b and ab are separable, and if a≠0 then a−1 is separable. The elements 0 and 1 lie in F and have linear minimal polynomials, so they are separable.

step 1.1algebra
3.1

The set Ks therefore satisfies the subfield criterion [L2] and contains F, so it is an intermediate field.

step 2.1L2∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 49 results over 21 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources