CorollaryStatement: 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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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

1.1

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 $K$ .

L1algebra

1.2

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

L1L2

2.1

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

step 1.1step 1.2L2∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 35 results over 10 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

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)