If $K$ and $K'$ are extensions of the same field $F$, is every isomorphism $K \to K'$ a $F$-isomorphism?

Question

If $K$ and $K'$ are extensions of the same field $F$, is every isomorphism $K \to K'$ a $F$-isomorphism, i.e., an isomorphism which restricts to the identity on $F$?

Answer 1

If there's a common subfield $\;L\neq F\;$ you can contradict that. For example, take

$$K=\Bbb Q(\sqrt[4]2)\;,\;\;K'=\Bbb Q(i\sqrt[4]2)\;,\;\;L=\Bbb Q(\sqrt2)\;,\;\;F=\Bbb Q$$

and take for example:

$$\phi:K\to K'\;,\;\;\phi(\sqrt[4]2)=i\sqrt[4]2$$

This extends naturally to a $\;\Bbb Q\,-$ isomorphism

$$\phi\left(a+b\sqrt[4]2+c(\sqrt[4]2)^2+d(\sqrt[4]2)^3\right):=a+bi\sqrt[4]2-c(\sqrt[4]2)^2-di(\sqrt[4]2)^3$$

Yet this isomorphism of fields is not a $\;\Bbb Q(\sqrt2)\,-$ isomorphism since

$$\phi\left(\sqrt2\right)=\phi(\sqrt[4]2)^2=\left(i\sqrt[4]2\right)^2=-\sqrt2\neq\sqrt2$$