I am trying to show that every algebraic extension of fields has a normal closure.
Let $L/K$ be an algebraic extension.
First suppose that the extension is finite. So $L=K(\alpha_1,...,\alpha_n)$ and let $f(X)=m_{\alpha_1}(X)\cdot\cdot\cdot m_{\alpha_n}(X)$ be the product of the minimal polynomials of $\alpha_i$ over $K$. Let $N$ be a splitting field of $f(X)$. Then $N\supseteq K$ and clearly $N$ is a normal closure of $L$.
I'm having problems with the case $[L:K]=\infty$. I tried using Zorn's lemma like this: I start by considering the collection $S$ of all intermediate fields $F$ such that $F/K$ has normal closure. Define the partial order $\leq$ on $S$ by the set inclusion $\subseteq$. Let $\mathcal C$ be any chain in $S$ and define the subfield $F'= \cup_{F\in \mathcal C}F$ of $L$. Then $F'$ is an upper bound for $\mathcal C$ in $S$ and so by Zorn's lemma, there is a maximal element $M$ in $S$.
Now I want to show that $M=L$. Suppose this is not true. Then $\exists \alpha \in L\backslash M$. Since $M(\alpha)/M$ is finite, by above, it has a normal closure $N_\alpha$ say.
Is it possible to show that $N_\alpha$ is normal over $K$?
Thank you for your help.