CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-11

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.

Over $F_{p}$ , $x^{p} - x$ and the zero polynomial induce the same function but are distinct polynomials

Statement refuted

Two formal polynomials over a field are equal whenever they induce the same function on that field.

Facts & Assumptions

Given: A prime $p$ , the field $F_{p} = Z / p$ , and the polynomials $f = x^{p} - x$ and $0$ in $F_{p} [x]$ .

[L1]

Evaluation substitutes an element into a formal polynomial, while the formal polynomial itself is its coefficient sequence (Evaluation and roots of a polynomial in a commutative target ring).

[L2]

Fermat's little theorem gives $a^{p} \equiv a (m o d p)$ for every integer $a$ (Fermat's little theorem: for prime $p$ , $p ∤ a$ implies $a^{p - 1} \equiv 1 (m o d p)$ , and always $a^{p} \equiv a (m o d p)$ ).

[L3]

The ring $Z / p$ is the quotient $Z / p Z$ (For every $n \in N$ , the congruence-class ring $Z / n$ is the quotient ring $Z / n Z$ ).

[L4]

A prime integer is greater than $1$ (Prime and composite integers: $p$ is prime when $p > 1$ and its only positive divisors are $1$ and $p$ ).

[L5]

For prime $p$ , the quotient ring $Z / p$ is a field (For every prime $p$ , the two operations on $Z / p$ make it a field).

Counterexample

technique · direct

1.1

Fact [L5] licenses the field $F_{p} = Z / p$ . For every $a \in F_{p}$ , choose an integer representative; [L2] and [L3] give $f (a) = a^{p} - a = 0$ , so $f$ and the zero polynomial induce the same function.

givenL1L2L3L5

2.1

Because [L4] gives $p > 1$ , the degrees $p$ and $1$ are distinct; as a coefficient sequence, $f$ has coefficient $1$ in degree $p$ and is therefore nonzero. Hence [L1] distinguishes $f$ from the zero formal polynomial and refutes the statement.

givenL1L3L4∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

36 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

Neil Donaldson, Math 120B Notes, Section 22, polynomial functions (standard reference, not scraped)

Over FpFp​, xp−xxp−x and the zero polynomial induce the same function but are distinct polynomials