I'm studying Keith Conrad's notes on infinite Galois theory (here) and I had difficulties proving the following theorem:
Theorem 3.2. For an algebraic extension $L/K$, the following properties are equivalent:
(1) $L = \bigcup_i L_i$, with each $L_i/K$ a finite Galois extension,
(2) $L$ is the splitting field over $K$ of a set of separable polynomials in $K[X]$,
(3) $L^{\mathrm{Aut}(L/K)} = K$,
(4) $L/K$ is both separable (every element of $L$ has a separable minimal polynomial over $K$) and normal (every irreducible polynomial in $K[X]$ that has a root in $L$ splits completely over $L$).
Proof. Exercise 3.2. $\square$
More specifically, my difficulty is in showing that any of the statements (1), (2), and (4) imply (3). In one of the appendices, Conrad presents the following result:
Theorem A.1. Let $L/K$ be a Galois extension and $\varphi: E \to L$ be a $K$-homomorphism of an intermediate field. There is an extension of $\varphi$ to a $K$-automorphism $L \to L$.
An immediate corollary (listed as Corollary A.2 just after the proof) is that if $F\subset L$ is a finite Galois subextension, then every element of $\text{Aut}(F/K)$ is a restriction of an element of $\text{Aut}(L/K)$. Using this result, I see how to prove statement (3) from any of the others. The issue is that the equivalence in Theorem 3.2 is used to define a Galois extension, while Theorem A.1 assumes that Galois extensions have already been defined. To avoid circular reasoning, I tried to find an argument that does not rely on Theorem A.1 (or, more specifically, the corollary), but I couldn't. I managed to prove that (3) implies (4). If I can show that (4) implies (3), then I only need to establish the equivalence of (1), (2), and (4) (which I know how to do). That said, my question is:
- How to show that (4) implies (3) without using Theorem A.1?
From what I saw in the proof of Theorem A.1, the hypothesis that $L/K$ is Galois is only used to ensure that if an irreducible polynomial $f(X)\in K[X]$ has a root in $L$, then it completely decomposes in $L[X]$—in other words, the normality property of the extension $L/K$ is used. Given the hypothesis of (4), I could use the same proof as Theorem A.1 to ensure that the result holds for normal extensions (instead of Galois extensions), and then use this to conclude that if $F\subset L$ is Galois and finite, then every element of $\text{Aut}(F/K)$ is a restriction of an element of $\text{Aut}(L/K)$. Using this fact, (3) follows easily by applying finite Galois theory to $F$. Still, I would like to know if anyone knows of an argument that shows (3) without using Theorem A.1 (or its adaptation for normal extensions).