If $E$ is an extension field of $F$, is $F$ necessarily (without assuming the axiom of choice) complemented as a vector subspace of $E$? (Of course the answer is easily yes if the extension is finite.)

The motivation for the question is pedagogical. I'm looking for as elementary as possible (hence no AC) a way of addressing the following basic linear algebra question: if a linear system $A x = b$, with $A$ an $m\times n$ matrix over $F$ and $b \in F^m$, has a solution $x \in E^n$, does it necessarily have a solution in $F^n$?