CounterexampleConstruction: AI-adaptedVerification: AI-adaptedPipeline-generatedjudge pass (gpt-5.6-terra)audited 2026-09-07

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.

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

1.1givenF3F4

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 $k$ .

2.1F1F2step 1.1algebra

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 $z = u + ϵ$ .

3.1step 1.1step 2.1algebra∎

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.

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)