Smith-Normalform

\mathbb{Q}[t]: \left(\begin{matrix} t^3-t & t^3-2t+1\\ -t3+t^2 & -t^3+t^2+t-1\\ -t^2+t & t^2+2t-1\end{matrix}\right) \cdot \begin{pmatrix}p_1\\p_2\end{pmatrix} = \begin{pmatrix}2t^3-t^2-1\\ -2t^3 +3t^2-2t+1\\ -t^2+1\end{pmatrix}$$​​ SNF:$$\begin{pmatrix} 0 & 0 & -1 \\ 2-t & 1-t & 1 \\ 1-t & -t & 1 \end{pmatrix} \begin{pmatrix} t^3-t & t^3-2t+1\\ -t^3+t^2 & -t^3+t^2+t-1\\ -t^2+t & t^2+2t-1\end{pmatrix} \begin{pmatrix} 1 & 1-t\\ -1 & t\end{pmatrix} = \begin{pmatrix} t-1 & 0 \\ 0 & t^2-t\\ 0 & 0 \end{pmatrix}$$​​