Which elements are fixed by the Galois group? If $\mathbb{C}$ is viewed as field extension of $\mathbb{R}$, then the elements of $\mathbb{C}$ that are invariant under the action of the Galois group are precisely the elements of $\mathbb{R}$. However, as far as I know, it's possible to have elements in the algebraic closure $\overline{K}$ of a field $K$ that are invariant under the action of the Galois group, despite not being elements of $K$.

Question. Is this possible, and if so, what would be an example?

Note that splitting fields cannot work as examples.
 A: If $L$ is any normal (but not necessarily finite) extension of a field $K$, then the fixed field of $Gal(L/K)$ is just the set $E$ of elements of $L$ whose minimal polynomial over $K$ has only one root.  Indeed, if $a\in E$, then any $g\in Gal(L/K)$ must send $a$ to another root of the same minimal polynomial so $g(a)=a$.  Conversely, if the minimal polynomial of $a$ has more than one root, then there is another root $b\in L$ since $L$ is normal.  There is then an embedding $K(a)\to L$ sending $a$ to $b$, and this embedding can be extended to an isomorphism $L\to L$ by transfinite induction, extending it to one new element at a time using normality of $L$ over $K$.
More explicitly, if $K$ has characteristic $0$, then this $E$ is just $K$, since the minimal polynomial of any $a\in L$ must have distinct roots.  If $K$ has characteristic $p>0$, then $E$ is the set of $a\in L$ such that $a^{p^n}\in K$ for some $n\in\mathbb{N}$.  Indeed, if $a^{p^n}\in K$, then the minimal polynomial of $a$ divides the polynomial $x^{p^n}-a^{p^n}=(x-a)^{p^n}$, and so has no roots besides $a$.  Conversely, if $a\in K$ has minimal polynomial $g(x)=f(x^{p^n})$ where $f\in K[x]$ is separable, $g$ has $\deg f$ distinct roots, so if $a\in E$ then $f$ is linear and so $g(a)=0$ implies $a^{p^n}\in K$.
In particular, if $K$ is any field of characteristic $p$ which is not closed under taking $p$th roots, then if $L=\overline{K}$, the fixed field of $Gal(L/K)$ is larger than $K$.  An explicit example of such a field is a field of rational functions $\mathbb{F}_p(t)$, in which the variable $t$ has no $p$th root.
If you want more details, this is all very standard Galois theory you can find in any good reference on the subject.  For instance, the main statement I made in the first two paragraphs is basically contained in Proposition V.6.11 of Lang's Algebra.
