Lemma 10.34.10. Suppose that $k$ is a field and suppose that $V$ is a nonzero vector space over $k$. Assume the dimension of $V$ (which is a cardinal number) is smaller than the cardinality of $k$. Then for any linear operator $T : V \to V$ there exists some monic polynomial $P(t) \in k[t]$ such that $P(T)$ is not invertible.

Proof. If not then $V$ inherits the structure of a vector space over the field $k(t)$. But the dimension of $k(t)$ over $k$ is at least the cardinality of $k$ for example due to the fact that the elements $\frac{1}{t - \lambda }$ are $k$-linearly independent. $\square$

Comment #4 by Johan on

Spelling: "Asssume".

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