= Square-free minimal polynomial criterion for polynomial kernel stabilization
{title2=$\ker p(T)^n=\ker p(T)\text{ for all }p,n\ge1$}
For a <linear operator> on a finite-dimensional <vector space> over a <field> $K$, all polynomial kernels stabilize at their first power exactly when its <minimal polynomial> $m$ has no repeated <irreducible polynomial> factor in $K[X]$. This formulation does not require those factors to be linear.
To prove it, the <Bézout's identity for polynomials> gives $\ker q(T)=\ker\gcd(q,m)(T)$: a greatest common divisor is a combination of $q$ and $m$, while $q$ is its multiple. If $m$ is square-free, $\gcd(p^n,m)=\gcd(p,m)$, proving the displayed equality. Conversely, if an irreducible factor $f$ occurs to exponent $e\ge2$, its <primary decomposition theorem> summand has <minimal polynomial> $f^e$. On this summand $f(T)$ is nonzero, while $f(T)^e=0$. Thus the kernels of those two powers differ. An <eigenbasis> follows from this criterion precisely when the distinct factors of $m$ are also linear over $K$; a real rotation through a right angle has square-free $m=X^2+1$ but no real <eigenvalues>.
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