False statementConstruction: AI-adaptedVerification: AI-adaptedSession-authored (Fable 5 assisted)precheck passjudge pass (gpt-5.6-terra)audited 2026-08-28

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False statement: every finite-dimensional algebra over a field is semisimple

Statement

False claim. Every finite-dimensional algebra over a field is semisimple.

Facts & Assumptions

Given: A prime $p$ and the group algebra $A = (Z / p) [C_{p}]$ .

[L1]

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

[L2]

If $G$ is finite, then $\dim_{k} k [G] = ∣ G ∣$ (If $G$ is finite then $\dim_{k} k [G] = ∣ G ∣$ ).

[L3]

If $char k ∣ ∣ G ∣$ , then $k [G]$ is not semisimple (If $char k ∣ ∣ G ∣$ , then $k [G]$ is not semisimple).

Refutation

technique · direct

1.1

By [L1], the coefficient ring $Z / p$ is a field, so $A$ is a finite-dimensional algebra over a field. By [L2], $\dim_{Z / p} A = ∣ C_{p} ∣ = p .$

L1L2givenalgebra

2.1

The field $Z / p$ has characteristic $p$ , and $∣ C_{p} ∣ = p$ , so [L3] applies and shows that $A$ is not semisimple. Therefore a finite-dimensional algebra over a field need not be semisimple.

L3step 1.1givenalgebra∎

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

16 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

Pavel Etingof et al., Introduction to Representation Theory, Proposition 3.2 (standard reference, not scraped)