Alphabeta Math
False statementConstruction: Literature-sourcedVerification: AI-adaptedprecheck passaudited 2026-08-16
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FALSE: If the minimal polynomial splits, then the endomorphism is diagonalisable

Statement

False claim. If the minimal polynomial of an endomorphism splits over the base field, then the endomorphism is diagonalisable.

Facts & Assumptions

Given: The matrix J=(1101).

[L1]

Diagonalisability requires the minimal polynomial to be a product of distinct linear factors (An endomorphism is diagonalisable if and only if its minimal polynomial is a product of distinct linear factors).

[L2]

Splitting permits repeated linear factors (Polynomials that split and splitting fields of a polynomial or a family of polynomials).

Refutation

technique · direct
1.1L2algebra

One has (J−I)2=0 but J−I≠0, so μJ=(x−1)2. This polynomial splits by [L2], but its root is repeated.

2.1step 1.1L1∎

By [L1], J is not diagonalisable. Thus splitting alone is insufficient; squarefreeness is the missing condition.

Depends on

Used by

Nothing in the library uses this result yet.

Dependency tree · two levels

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Sources