Let $F$ be an extension of $K$ (they are both fields). I know that if $F$ has finite degree over $K$, then the following things are equivalent:
1) $F$ is such that every irreducible polynomial in $K[x]$ has a root in $F$
2) If $T$ is an extension of $F$ and $\alpha$ an automorphism of $T$ such that $\alpha(b)=b$ for each $b\in K$, then $\alpha(F)=F$
The implications are true if $F$ is an algebraic extension of $K$. Do they hold also in the general case?