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.

  • 2
    $\begingroup$ 1. Why is your $F'$ in $S$? 2. It is false that in a tower of algebraic extensions $N/M/K$ that $N/M$ and $M/K$ being normal implies $N/K$ is normal. There are already very simple counterexamples taking $K = \mathbf Q$ and $[N:M] = [M:K] = 2$, which you should be able to find on your own (if you know examples of non-normal extensions). $\endgroup$ – KCd Aug 3 '15 at 18:43
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    $\begingroup$ Can't you just take the intersection of extensions of $L$ normal over $K$? $\endgroup$ – neth Aug 3 '15 at 18:47
  • $\begingroup$ @KCd Would a counterexample be $N=\mathbb Q (\sqrt[4]{2})$, $M=\mathbb Q(\sqrt{2})$? Is there a way to prove the general case using Zorn's lemma? $\endgroup$ – eddie Aug 3 '15 at 20:00
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    $\begingroup$ Try looking at the proof that fields have algebraic closures, which uses Zorn's lemma too. $\endgroup$ – KCd Aug 3 '15 at 20:01
  • $\begingroup$ @neth sorry for the silly question but is the intersection of any collection of normal extensions of $K$ normal over $K$ (I know that it's true for a finite collection)? $\endgroup$ – eddie Aug 3 '15 at 20:08

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