TheoremStatement: Literature-sourcedProof: AI-adaptedprecheck passaudited 2026-08-17

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.

Pure inseparability is transitive in towers and stable under composita

Statement

If $F \subseteq K \subseteq L$ and both $K / F$ and $L / K$ are purely inseparable, then $L / F$ is purely inseparable. If $K_{1} / F$ and $K_{2} / F$ are purely inseparable subextensions of a common algebraic extension, then their compositum $K_{1} K_{2} / F$ is purely inseparable.

Facts & Assumptions

Given: Purely inseparable extensions in one of the configurations of the Statement.

[L1]

In characteristic $p$ , pure inseparability is equivalent to the elementwise condition that a suitable $p$ -power lies in the base (Pure inseparability and its conjugate, embedding, and separable-degree criteria).

[L2]

Frobenius respects addition, multiplication, and nonzero inverses in characteristic $p$ (Frobenius $x \mapsto x^{p}$ is an injective endomorphism in characteristic $p$ , and an automorphism for finite fields).

[L3]

A compositum is the subfield generated by the two subextensions, so each of its elements lies in a subfield generated by finitely many elements from them (Field extensions, generated subrings $F [S]$ , generated subfields $F (S)$ , and simple extensions).

Proof

technique · direct

1.1L1algebra

For $a \in L$ , choose $m$ with $a^{p^{m}} \in K$ and then $n$ with $(a^{p^{m}})^{p^{n}} \in F$ using [L1]. Thus $a^{p^{m + n}} \in F$ , so $L / F$ is purely inseparable.

1.2L1L2L3

For $a \in K_{1} K_{2}$ , [L3] places $a$ in $F (u_{1}, \dots, u_{r}, v_{1}, \dots, v_{s})$ with $u_{i} \in K_{1}$ and $v_{j} \in K_{2}$ . Choose one exponent $N$ whose $p^{N}$ th power sends every generator into $F$ . Applying Frobenius to a rational expression for $a$ and using [L2] gives $a^{p^{N}} \in F$ .

2.1step 1.1step 1.2L1∎

Hence the compositum is purely inseparable by [L1]. In characteristic zero all extensions in the hypotheses are trivial, so both conclusions hold there as well.

Depends on

Used by

Separable generation after finite purely inseparable extensions Lemma

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

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)