Normal Subgroup of Galois Group

Let $$L/K$$ be a Galois extension, and let $$R\subseteq L$$ be a subring such that $$\tau(R)=R$$ for every $$\tau\in\text{Gal}(L/K)$$.

Let $$\alpha\in R$$. How would I show that $$H=\{\tau\in\text{Gal}(L/K):\tau(\alpha)=\alpha\}$$ a normal subgroup of $$\text{Gal}(L/K)$$?

I've tried using the Fundamental Theorem of Galois Theory, by which it would be equivalent to show that $$L^H/K$$ is a Galois extension. I would think that $$L^H=K(\alpha)$$, but haven't been able to show this, or that assuming this the extension is Galois.

• What do you think? – ÍgjøgnumMeg Jan 27 at 11:37
• Apologies for the phrasing, I've been trying to follow a proof which assumes that $H$ is normal and haven't been able to show this either way myself yet. Based on the provenance I think it is true, I'll rephrase the question – Dave Jan 27 at 11:45

Unless I misunderstood something the claim is false. Take the field extension $$\mathbb{Q} \subset \mathbb{Q}(\sqrt{2},i)$$, which is a Galois extension, being the splitting field of $$x^4-2$$
Now take $$R = \mathbb{Q}(\sqrt{2},i)$$ and obviously the condition is satisfied. Now pick $$\alpha = \sqrt{2}$$. Then it's not hard to show that $$L^H = \mathbb{Q}(\sqrt{2})$$. But this is obviously not a Galois extension of $$\mathbb{Q}$$, so H can't be a normal group of $$\text{Gal}(L/K)$$
As you have mentioned it's true that $$L^H = K(\alpha)$$. To prove this first note that $$K(\alpha) \subseteq L^H$$, as obviously $$K(\alpha)$$ is fixed pointwise by $$H$$. For the other inclusion, by the Fundamental Theorem of Galois Theory we have $$K(\alpha) = L^{H'}$$ for some subgroup $$H'$$ of $$\text{Gal}(L/K)$$. Now as $$H'$$ fixes $$\alpha$$ we have that $$H' \le H$$. From this $$L^{H} \subseteq L^{H'} = K(\alpha)$$. Hence $$L^H = K(\alpha)$$.
Furthermore, the reason why the statement fails is because $$R$$ isn't necessarily a subring of $$L^H$$. This means that although the conjugates of $$\alpha$$ are in $$R$$, they might not be in $$K(\alpha)$$, which has to be the case if it were a Galois extension of $$K$$.