Lemma 11.6.2. Let $A$ be a finite central simple $k$-algebra. Any automorphism of $A$ is inner. In particular, any automorphism of $\text{Mat}(n \times n, k)$ is inner.

**Proof.**
Note that $A$ is a finite central simple algebra over the center of $A$ which is a finite field extension of $k$, see Lemma 11.4.2. Hence the Skolem-Noether theorem (Theorem 11.6.1) applies.
$\square$

## 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Your email address will not be published.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 (1)

Comment #8883 by Awllower on

There are also: