= Diagonalization of a cyclic shift over a field
{title2=$S\text{ diagonalizable over }K\iff X^n-1\text{ splits into distinct linear factors over }K$}
Let $S$ send successive standard <basis vectors> of $K^n$ to the next one, wrapping the last to the first. Its <minimal polynomial> is $X^n-1$: this polynomial annihilates it, and the cyclic vectors $e_1,Se_1,\ldots,S^{n-1}e_1$ form a <basis>, excluding any annihilator of smaller degree. The <diagonalizability criterion using the minimal polynomial> gives the displayed equivalence. A splitting polynomial is square-free exactly when the <characteristic of a field> does not divide $n$, as follows by taking its <derivative> $nX^{n-1}$ and checking common roots. Splitting over the specified field is a separate requirement. For $n=3$, the shift is diagonalizable over the <complex numbers> and $\mathbb F_7$, but not over $\mathbb F_2$ or $\mathbb F_3$.
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