Element of group ring not invertible, but invertible in bigger group ring.

Question

I have thought about such problems:

• Is there a field $F$ and a group $G$ such that there exists element $x$ of group ring $FG$ which is not invertible in $FG$, but is invertible in $KG$, where $K$ is some field extension of $F$?
• A more particular version of my question is this: is it known for some $F$ being not algebraically closed and $K$ being algebraic closure of $F$?