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.
$\square$

## Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like `$\pi$`

). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

## Comments (2)

Comment #4 by Johan on

Comment #16 by Johan on

There are also: