CounterexampleConstruction: Literature-sourcedVerification: AI-adaptedprecheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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.

There is no field with six elements

Statement refuted

There exists a field with six elements.

Facts & Assumptions

Given: A hypothetical field $F$ with $∣ F ∣ = 6$ .

[L1]

Every finite field has order $p^{n}$ for a prime $p$ and a positive integer $n$ (Every finite field has order $p^{n}$ for a unique prime characteristic $p$ and positive integer $n$ ).

[L2]

A prime greater than $1$ has no nontrivial factorization (Prime and composite integers: $p$ is prime when $p > 1$ and its only positive divisors are $1$ and $p$ ).

[L3]

If a prime divides a product, it divides one of the factors (Euclid's lemma: if $p$ is prime and $p ∣ a b$ then $p ∣ a$ or $p ∣ b$ ).

Counterexample

technique · contradiction

1.1givenL1assume-contra

Suppose, for contradiction, that $F$ exists. By [L1], $6 = p^{n}$ for some prime $p$ and $n \geq 1$ .

2.1step 1.1L2L3algebra

The prime $p$ divides $6 = 2 \cdot 3$ , so [L3] gives $p ∣ 2$ or $p ∣ 3$ , and [L2] forces $p = 2$ or $p = 3$ . But no positive power of $2$ is $6$ , and no positive power of $3$ is $6$ : at exponent one the values are $2$ and $3$ , while at exponent at least two they are divisible by $4$ or $9$ .

3.1step 2.1discharge-contradiction∎

This contradiction proves that no six-element field exists.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

29 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

K. Conrad, Finite Fields, Section 1 (standard reference, not scraped)