Automorphisms of $GL_n(K)$, and $SL_n(K)$ There are (at least) two types of "trivial" automorphisms of $GL_n(K)$: the inner automorphisms, and the automorphisms arising as extensions of automorphisms of $K$.("trivial" in that they can be thought of for any field $K$)
However, both of these types of automorphism fix $SL_n(K)$.  I was looking for examples of a field $K$ and an automorphism of $GL_n(K)$ that doesn't fix $SL_n(K)$.
Are there any ? Are there some for any field $K\neq \mathbb{F}_2$ (obviously this field doesn't have any since $GL_n(\mathbb{F}_2) = SL_n(\mathbb{F}_2)$) ? If not, for which fields can there exist some ? Are there nice and easy examples ? 

In general, the derived subgroup of $ GL_n(K) $ is $ SL_n(K) $ as long as $ |K| \geq 3 $ or $ n \geq 3 $. If $ |K| = 2 $, the claim is trivially true; and if $ |K| \geq 3 $, this result implies that any automorphism fixes $ SL_n(K) $ as automorphisms map commutators to commutators.
