= Polynomial substitution in a size-two Jordan block
{title2=$p(J_2(\lambda))=p(\lambda)I+p'(\lambda)N$}
For $T=\lambda I+N$ with $N^2=0\ne N$ and a <polynomial> $p$, the binomial formula gives $p(T)=p(\lambda)I+p'(\lambda)N$. Hence its <module> on the two-dimensional space is indecomposable exactly when $p'(\lambda)\ne0$: in that case the nonzero nilpotent part is one <Jordan block>; otherwise the operator is scalar and every line is a <submodule>. For $p(X)=X^2$ over a <field> of characteristic different from two, the criterion is $\lambda\ne0$.
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