Is $L$ normal over $E$ whenever it is normal over a subfield of $E$? Let $L$ be a normal extension, with infinite degree, of a field $K$, and let $E$ be a subfield of $L$ containing $K$.
I'd like to see either a proof or a counterexample of the following:

$L$ is a normal extension of $E$.

(I know of a proof of the analogous result for the case where $[L:K]$ is finite, but it does not extend to the case of infinite $[L:K]$.)

EDIT: @Lubin: FWIW, the proof I alluded to is not mine; it's from a solved exercise in John M. Howie's Fields and Galois theory, p. 212 (solution to exercise 7.7).  This proof/solution is very short:

[solution]
  7.7. Since $L$ is a normal extension of $K$, it is a splitting field for some polynomial $f$ in $K[X]$.  Since $f \in E[X]$, we conclude that $L$ is a normal extension of $E$.

The last conclusion follows for a theorem stated on p. 103, namely:

Theorem 7.13
  A finite extension of $L$ of a field $K$ is normal if and only if it is a splitting field for some polynomial in $K[X]$.

It is worth noting that the solution the textbook gives for exercise 7.7 (and quoted in its entirety above) is only partial, since it relies on a theorem (7.13) having the premise that $[L:K]$ is finite, whereas the original statement for exercise 7.7 places no such condition on $[L:K]$.  
For the sake of completeness, here's the full statement of exercise 7.7 (from p. 109):

7.7.  Let $L$ be a normal extension of a field $K$, and let $E$ be a subfield of $L$ containing $K$.  Show that $L$ is a normal extension of $E$.

(The first sentence of the problem statement is identical to the first sentence of my original post, except that it says nothing about the degree of $L:K$.)
 A: Sorry for my careless reading of your question. Here’s the proof in the general case:
The difficulty with generalizing the proof you have seen is that “finite and normal $\Leftrightarrow$ splitting field” is not useful for infinite extensions. You need a useful concept of normality that works for arbitrary algebraic extensions, and here it is:
The following are equivalent statements about an algebraic extension $L\supset K$:
$\quad$1. $L$ is normal over $K$;
$\quad$2. Whenever $\Omega$ is an algebraically closed field containing $L$, every $K$-homomorphism of $L$ into $\Omega$ sends $L$ into itself.
I don’t know what definition of normality you have been working with, but I think you can see easily that for finite extensions, $L$ is a splitting field of a $K$-polynomial if and only if $L\supset K$ satisfies condition #2 above.
Now suppose that $E$ is an intermediate field of a normal extension $L\supset K$. Then clearly, by #2, $L$ is normal over $E$ as well, ’cause any $E$-homomorphism of $L$ into $\Omega$ is a fortiori a $K$-homomorphism of $L$ into $\Omega$.
