# Extensions of fields with lots of symmetry

[Cute question heard elsewhere]

Is there a nice characterization of extensions of fields $K/k$ such that whenever $E/k$ and $E'/k$ are subextensions and $\sigma:E\to E'$ is an isomorphism over $k$, there is a $\tilde\sigma\in\mathrm{Aut}(K/k)$ such that $\tilde\sigma|_E=\sigma$?

Normal extensions and those without proper subextensions have that property. On the other hand, $\mathbb Q(\sqrt{2})/\mathbb Q$ doesn't.

• I always thought that the only subextension of Q(2^{1/4})/Q was Q(2^{1/2}). Am I wrong? – Kevin Buzzard Feb 26 '10 at 14:33
• $E$ can coincide with $E'$. – Mariano Suárez-Álvarez Feb 26 '10 at 14:39
• Touch\'e ;-) – Kevin Buzzard Feb 26 '10 at 14:46

Take for $k$ an algebraically closed field and for $K$ any algebraically closed extension of transcendence degree one.Then if you give yourself a $k$- morphism $\sigma :E \to E'$ it extends to an endomorphism $\tilde \sigma : K \to K$, since $K$ is algebraic over $E$ and algebraically closed. This extended morphism is surjective, hence an automorphism, because $\tilde \sigma (K) \subset K$ is algebraic and $\tilde \sigma (K)$ is algebraically closed. Since $k$ is fixed under $\tilde \sigma$ you get your result.
• Why is $K$ algebraic over $E$? – Mariano Suárez-Álvarez Feb 26 '10 at 15:13
• Ah. I'd missed the fact that $k$ is algebraically closed. Thanks! – Mariano Suárez-Álvarez Feb 26 '10 at 16:52