Does an automorphism fix some elements?

Question

Let $K / F$ be an extension of fields (it implies that $K$ is a vector space over $F$). As an automorphism of $K$ is an isomorphism from $K$ to $K$ (isomorphism of vector spaces), is true that $\sigma(\alpha) = \alpha$ for all $\alpha \in F$ and for all automorphism $\sigma$ of $K$? My attemps have no result. Thank you so much.

Answer 1

Using your definition of "automorphism" as just an isomorphism of vector spaces over $F$, then this is false.

For example, $\mathbb{C}/\mathbb{R}$ is an extension of fields, but there are many vector space isomorphisms of $\mathbb{C}$ over $\mathbb{R}$ which do not fix the elements of $\mathbb{R}$. For example, $f(a+bi) = b+ai$.

Answer 2

The answer is "no". Take, for instance, the extension $$ \Bbb Q(i, \sqrt 2)/\Bbb Q(i) $$ Then there is an automorphism $\sigma$ of $\Bbb Q(i, \sqrt2)$ given by $$ \sigma(a + bi + c\sqrt2 + di\sqrt2) = a-bi+c\sqrt2 - di\sqrt2 $$ for $a, b, c, d\in \Bbb Q$. This automorphism clearly doesn't fix the elements of $\Bbb Q(i)$.