CorollaryStatement: Literature-sourcedProof: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + claude-sonnet-5)audited 2026-08-17

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Every purely inseparable algebraic extension is normal

Statement

Every purely inseparable algebraic extension is normal.

Facts & Assumptions

Given: A purely inseparable algebraic extension $K / F$ and an element $α \in K$ .

[L1]

In positive characteristic, the minimal polynomial of $α$ has the form $x^{p^{e}} - a$ ; in characteristic zero the extension is trivial (Pure inseparability and its conjugate, embedding, and separable-degree criteria).

[L2]

An algebraic extension is normal exactly when the minimal polynomial over the base of every one of its elements splits in the extension (A normal algebraic extension is one in which every minimal polynomial with a root in the extension splits there).

Proof

technique · direct

1.1L1algebra

In characteristic $p$ , [L1] gives $m_{α} (x) = x^{p^{e}} - a = (x - α)^{p^{e}}$ in $K [x]$ , so the minimal polynomial splits in $K$ .

1.2L1L2

In characteristic zero, [L1] gives $K = F$ , which is normal.

2.1step 1.1step 1.2L2∎

Thus every minimal polynomial required by [L2] splits in $K$ , and $K / F$ is normal.

Depends on

Used by

Fₚ(t)/Fₚ(tᵖ) is normal and inseparable with trivial automorphism group Counterexample

Dependency tree · two levels

13 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, Chapter 6 (standard reference, not scraped)
The Stacks Project, Section 9.15: Normal extensions (standard reference, not scraped)