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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An algebraic extension is purely inseparable over its separable closure

Statement

Let $K / F$ be algebraic and let $K_{s}$ be the separable closure of $F$ in $K$ . Then $K / K_{s}$ is purely inseparable.

Facts & Assumptions

Given: An algebraic extension $K / F$ , its separable closure $K_{s}$ , and an element $α \in K$ .

[L1]

The field $K_{s}$ consists exactly of the elements of $K$ separable over $F$ (The separable closure of the base inside an algebraic extension).

[L2]

In characteristic $p$ , the minimal polynomial of $α$ has a unique form $g (x^{p^{e}})$ with $g$ irreducible and separable (In characteristic $p$ , every irreducible polynomial is uniquely $g (x^{p^{e}})$ with $g$ irreducible and separable).

[L3]

Pure inseparability is the elementwise $p$ -power condition, with only the trivial case in characteristic zero (Purely inseparable algebraic extensions).

[L4]

Every field of characteristic zero is perfect (A field is perfect exactly when it has characteristic zero or its Frobenius map is surjective).

[L5]

In a perfect field, every nonconstant irreducible polynomial is separable (Perfect fields: every irreducible polynomial is separable).

Proof

technique · direct

1.1

Suppose $char F = p > 0$ and write the minimal polynomial of $α$ as $g (x^{p^{e}})$ using [L2]. Then $g (α^{p^{e}}) = 0$ , and the minimal polynomial over $F$ of $α^{p^{e}}$ divides the separable polynomial $g$ , so $α^{p^{e}}$ is separable over $F$ and belongs to $K_{s}$ by [L1].

L1L2algebra

2.1

Thus every element of $K$ has a suitable $p$ -power in $K_{s}$ , so [L3] makes $K / K_{s}$ purely inseparable.

step 1.1L3

3.1

In characteristic zero [L4] and [L5] make every irreducible polynomial separable, so $K_{s} = K$ by [L1]; the extension $K / K_{s}$ is trivial and purely inseparable by [L3].

L1L3L4L5∎

Depends on

Used by

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 34 results over 9 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

P. L. Clark, Field Theory, Chapters 4 and 5 (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, Chapters 3 and 5 (standard reference, not scraped)