The "extending the automorphism" theorem in Galois theory I recently heard that the following statement is a theorem in Galois theory.
Theorem.  Let $F$ be a field and $L/K/F$ be a tower of field extensions of $F$.  Suppose that $L/F$ and $K/F$ are both Galois extensions.  Then for any $\sigma \in \text{Gal(K/F)}$, there exists  $\tilde{\sigma} \in \text{Gal}(L/F)$ such that
\begin{equation*}
\tilde{\sigma}|_{K} = \sigma
\end{equation*}
I have a couple of questions:


*

*Is this theorem (as I have stated it) true?  

*Is the $\tilde{\sigma}$ unique?

*Do you know of an online reference (lecture notes, wikipedia, etc.) that states this result?  I would like to read the proof, see the context, etc.
Thanks so much!
 A: *

*Yes, the theorem is true as stated.

*No, in fact, it will never be unique if the inclusion $ K \subset L $ is proper, since you have a surjective (by this result) group homomorphism $ \textrm{Gal}(L/F) \to \textrm{Gal}(K/F) $ given by restriction to $ K $. By order comparisons, it follows that it has nontrivial kernel iff $ K \subset L $ is a proper inclusion, thus the fibers have more than one element.

*This result follows from the fact that if $ \phi : K_1 \to K_2 $ is an isomorphism of fields and $ f \in K_1[X] $ is a polynomial, then the isomorphism extends to an isomorphism $ \bar \phi : L_1 \to L_2 $ of splitting fields of $ f $ and $ \phi(f) $ respectively. Since a finite Galois extension of $ K $ is always the splitting field of some polynomial with coefficients in $ K $, we may take $ K_1 = K_2 = K $ and get an automorphism $ \bar \phi : L \to L $ extending $ \phi $. I don't have any specific references, but this result is proved in pretty much any book on Galois theory.

