DefinitionDefinition: AI-adaptedProof: Not applicablejudge pass (deepseek-v4-pro + claude-sonnet-5)audited 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.

$p$ -bases for finite exponent-one purely inseparable extensions

Definition

Let $K / F$ be a finite purely inseparable extension of characteristic $p > 0$ (Purely inseparable algebraic extensions) and suppose it has exponent at most one, meaning $a^{p} \in F$ for every $a \in K$ . A finite ordered family $(b_{1}, \dots, b_{r})$ in $K$ is a $p$ -basis of $K / F$ when the restricted monomials

$b_{1}^{e_{1}} \dots b_{r}^{e_{r}} (0 \leq e_{i} < p)$

form an $F$ -basis of $K$ (Basis of a vector space: a linearly independent spanning subset; and ordered basis: an injective finite list whose image is a basis). For $r = 0$ , the sole restricted monomial is the empty product $1$ , so the definition gives the basis $(1)$ of the trivial extension and degree one (The degree $[K : F] = \dim_{F} K$ of a finite field extension).

Depends on

Used by

A minimal generating family in a finite exponent-one purely inseparable extension is a p-basis and gives degree pʳ Theorem

Dependency tree · two levels

18 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, Chapter 5 (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, Chapter 3 (standard reference, not scraped)

pp-bases for finite exponent-one purely inseparable extensions