= Semisimple linear operator
A <linear operator> is semisimple if it becomes <diagonalizable> over an algebraic closure; equivalently, its <minimal polynomial> has no repeated root over that closure. If an invertible operator $A$ has finite order $r$ prime to the <characteristic of a field>, its <minimal polynomial> divides $X^r-1$. The derivative $rX^{r-1}$ is coprime to this polynomial, so $A$ is semisimple. Its invariant irreducible summands over a <finite field> become multiplication by scalars in suitable <finite field extensions>.
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