CounterexampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07‡ rests on unproved material (inherited)

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.

‡ Rests on 3 statements not proved in this library, by way of the results it cites. This item cites no such statement directly; it depends on results that do. The unproved premises it inherits are ‡ Cohen's first model: an infinite Dedekind-finite set of reals, ‡ Sierpiński 1947: the generalised continuum hypothesis implies the Axiom of Choice and ‡ The continuum hypothesis and its generalisation are independent of ZFC. Each is recorded with a citation to the literature and is not established here, because the track that would prove it has not yet been developed in this library. Everything else in this proof is proved here.

A reduced field spectrum becomes nonreduced

Statement refuted

False claim: reducedness survives algebraic extension of the ground field. Let $p$ be prime, $k = F_{p} (v)$ with $v$ transcendental, and $L = k [u] / (u^{p} - v)$ . Then $L$ is a field, but $L \otimes_{k} L ≅ L [ϵ] / (ϵ^{p}),$ so the base change of the reduced $k$ -scheme $Spec L$ along the finite algebraic extension $L / k$ is nonreduced.

Facts & Assumptions

Given: The objects, hypotheses and conventions in the statement above.

[F1]

For a field extension $K / k$ and a $k$ -scheme $X$ , the inverse image of every affine open $U = Spec A$ in $X_{K}$ is $Spec (A \otimes_{k} K)$ . These affine charts cover $X_{K}$ and are compatible on overlaps and with coefficient localizations. (Affine charts after extension of the ground field)

[F2]

Let $A \to C$ be a unital ring map. For any set of variables $(t_{i})$ and any ideal $I \subseteq A [t_{i}]$ , $(A [t_{i}] / I) \otimes_{A} C ≅ C [t_{i}] / I C [t_{i}] .$ Here the extended ideal is generated by the coefficient images of all elements of $I$ . For a multiplicative subset $M \subseteq A$ , $(M^{- 1} A) \otimes_{A} C ≅ {\overline{M}}^{- 1} C .$ These are ring isomorphisms; no flatness, finite-generation or nonzero-ring hypothesis is required. (Presentations and localization under base extension)

[F3]

Let $F$ have characteristic $p > 0$ , let $a \in F$ not be a $p$ th power in $F$ , and let $n \geq 1$ . Then $x^{p^{n}} - a$ is irreducible in $F [x]$ . (If $a$ is not a $p$ th power in a characteristic- $p$ field, then $x^{p^{n}} - a$ is irreducible for every $n \geq 1$ )

[F4]

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

Counterexample

1.1

By F4, $F_{p} [v]$ is a UFD. If $v = (a / b)^{p}$ for nonzero polynomials $a, b$ , comparison of the exponent of the prime polynomial $v$ in $v b^{p} = a^{p}$ gives $1 + p {ord}_{v} (b) = p {ord}_{v} (a)$ , impossible modulo $p$ . Thus $v$ is not a $p$ th power in $k$ , and F3 with exponent parameter 1 proves $u^{p} - v$ irreducible. Its quotient $L$ is a field of degree $p$ over $k$ .

givenF3F4

2.1

By F1 and F2 the base-change ring is $L [z] / (z^{p} - v)$ . In $L$ one has $v = u^{p}$ , and the characteristic- $p$ binomial formula gives $z^{p} - u^{p} = (z - u)^{p}$ . Substitution $ϵ = z - u$ supplies the claimed ring isomorphism, with inverse $z = u + ϵ$ .

F1F2step 1.1algebra

3.1

The classes $1, ϵ, \dots, ϵ^{p - 1}$ form a basis by division by the monic polynomial $ϵ^{p}$ . Since $p \geq 2$ , $ϵ$ is nonzero but nilpotent. Thus this ring, whose unique prime is $(ϵ)$ , is nonreduced whereas $L$ is reduced. The argument includes $p = 2$ ; no integer-only Eisenstein criterion is used.

step 1.1step 2.1algebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

17 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

Vakil 10.4.G (standard reference, not scraped)