False statementConstruction: AI-adaptedVerification: AI-generatedSession-authored (Fable 5 assisted)precheck passjudge pass (deepseek-v4-pro + gpt-5.6-terra)audited 2026-08-16

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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.
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FALSE: $F_{p^{n}}$ is the ring $Z / p^{n} Z$

Statement

For every prime $p$ and positive integer $n$ , the finite field $F_{p^{n}}$ is the quotient ring $Z / p^{n} Z$ .

Facts & Assumptions

Given: The case $p = 2$ , $n = 2$ .

[L1]

A field with four elements exists (For every prime $p$ and $n \geq 1$ , a field with $p^{n}$ elements exists).

[L2]

The congruence-class ring $Z / 4$ is the quotient ring $Z / 4 Z$ (For every $n \in N$ , the congruence-class ring $Z / n$ is the quotient ring $Z / n Z$ ).

[L3]

In a field every nonzero element has an inverse, so a nonzero nilpotent cannot exist (Field).

Refutation

technique · direct

1.1

By [L1], $F_{4}$ is a field. In $Z / 4 Z$ , the class $[2]$ is nonzero but $[2]^{2} = [4] = [0]$ .

givenL1L2algebra

2.1

Thus $Z / 4 Z$ has a nonzero nilpotent and is not a field by [L3], so it cannot be isomorphic to $F_{4}$ .

step 1.1L3

3.1

This single case refutes the universal identification. Equal cardinality does not determine a ring structure.

step 2.1∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · next 3 levels

Direct dependencies and their dependencies through the next three levels: 53 results over 11 levels. An arrow runs from a result to what uses it, and this result sits at the bottom with a heavier outline. Click the chart to enlarge it.

Sources

K. Conrad, Finite Fields, Remarks 1.9 and 2.4 (standard reference, not scraped)

FALSE: FpnFpn​ is the ring Z/pnZZ/pnZ