Example of a non-Galois extension with $[L : K] = |Aut(L/K)|$ When an extension of fields $L/K$ is finite, we always have $|\operatorname{Aut}(L/K)| \leq [L : K]$, and if $L/K$ is Galois then $|\operatorname{Aut}(L/K)| = [L : K]$.  Is the converse true?
Is there an example of an infinite extension $L/K$ with $|\operatorname{Aut}(L/K)| > [L : K]$?  

This addresses your second question.
Take $K$ to be the field with a prime number $p$ of elements, and $L$ its algebraic closure. 
Then $\operatorname{Gal}(L/K)$ is uncontable (it is isomorphic to the profinite closure of the integers).
However, the degree of $L$ over $K$ is countable, as $L$ is the union (direct limit) of all fields of order $p^{n}$, for $n$ in the positive integers. (Actually, $L$ itself is countable.)
