A generating set for the kernel of a given module homomorphism

Let V, T, A, et cetera be as in this previous exercise.

Show that \mathsf{ker}\ \varphi is generated by D = \{\omega_1,\ldots,\omega_n\}.

Let \zeta \in \mathsf{ker}\ \varphi. Now \varphi(\zeta) = 0, and by this previous exercise, we have \zeta \in \mathsf{span}_{F[x]}(\omega_1,\ldots,\omega_n) + \mathsf{span}_F(\xi_1,\ldots,\xi_n). Recall that the \omega_i are in \mathsf{ker}\ \varphi, so that 0 = \varphi(\zeta) = \sum b_i v_i for some b_i \in F. Since the v_i are a basis for V over F, we have b_i = 0 for all i. Thus \mathsf{ker}\ \varphi \subseteq \mathsf{span}_{F[x]}(\omega_1,\ldots,\omega_n). The reverse inclusion is immediate, and so the \omega_i are a generating set for \mathsf{ker}\ \varphi over F[x].

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