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- Jan 26, 2012

- 995

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**Problem**: Let $V$ be a finite-dimensional $K[X]$-module, and let $\phi$ be the associated operator on $V$. Suppose that $\Delta$ represents $\phi$ with respect to some basis. Prove that if $\Delta$ is a diagonal matrix (no nonzero entries off the diagonal), and the diagonal entries of $\Delta$ are pairwise distinct, then $V$ is a cyclic $K[X]$-module.

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