ExampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge 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.

$F_{p} (t) / F_{p} (t^{p})$ is purely inseparable of degree $p$ and separable degree one

Example

For a prime $p$ , the extension $F_{p} (t) / F_{p} (t^{p})$ is purely inseparable, has ordinary degree $p$ , and has separable degree one.

Facts & Assumptions

Given: A prime $p$ , the field $K = F_{p} (t)$ , and its subfield $F = F_{p} (t^{p})$ .

[L1]

A rational function field is the fraction field of its polynomial ring (For a field $F$ , $F (t) = Frac (F [t])$ is its rational function field; in particular $R (t) = Frac (R [t])$ ).

[L2]

A polynomial ring over a field is a unique factorization domain (For every field $F$ , $F [x]$ is a unique factorisation domain).

[L3]

If a constant is not a $p$ th power, then $x^{p} - a$ is irreducible (If $a$ is not a $p$ th power in a characteristic- $p$ field, then $x^{p^{n}} - a$ is irreducible for every $n \geq 1$ ).

[L4]

A simple extension has the power basis and degree of its minimal polynomial (A simple algebraic extension is its minimal-polynomial quotient and has power basis $1, a, \dots, a^{n - 1}$ and degree $n$ ).

[L5]

A finite purely inseparable extension has separable degree one (Pure inseparability and its conjugate, embedding, and separable-degree criteria).

Verification

technique · direct

1.1L1L2algebra

Put $u = t^{p}$ , so $F = F_{p} (u)$ . If $u = (r (u) / s (u))^{p}$ for nonzero coprime $r, s \in F_{p} [u]$ , then unique factorization gives $1 + p v_{u} (s) = p v_{u} (r)$ , which is impossible modulo $p$ . Thus $u$ is not a $p$ th power in $F$ .

2.1step 1.1L3L4

The element $t$ is a root of $x^{p} - u$ , which is irreducible by [L3]. Hence [L4] gives $K = F (t)$ and $[K : F] = p$ .

3.1L5algebra∎

For every $z = r (t) / s (t) \in K$ , characteristic $p$ gives $z^{p} = r (t)^{p} / s (t)^{p} \in F_{p} (t^{p}) = F$ . Thus $K / F$ is purely inseparable, and [L5] gives $[K : F]_{s} = 1$ .

Depends on

Used by

Global functions on geometrically connected and geometrically reduced proper schemes Corollary
FALSE: every finite extension satisfies [K:F]ₛ=[K:F] False statement

Dependency tree · two levels

28 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 4 (standard reference, not scraped)
J. S. Milne, Fields and Galois Theory, Chapters 3 and 5 (standard reference, not scraped)

Fp(t)/Fp(tp)Fp​(t)/Fp​(tp) is purely inseparable of degree pp and separable degree one