I am trying to show that $\mathbb{Q}(X)/\mathbb{Q}$ is Galois where $X = \{ \sqrt{p}: p \text{ prime}\}$. This is the same exercise that can be found here: Let $X = \{ \sqrt{p} : p \text{ is prime} \}$, $Y \subseteq X$ and $\sqrt{p} \not\in Y$. Show that $[\mathbb{Q}(Y)(\sqrt{p}) : \mathbb{Q}(Y)] = 2$.. Specifically, I do not sure how to complete part (b) which asks to show that If $Y \subset X$, then there is $\sigma \in \text{Gal}(\mathbb{Q}(X)/ \mathbb{Q})$. The hint is to use Zorn's Lemma and I am not sure I am applying it correctly: (Note that some of the notation I am using comes from the linked post).
I let $\mathcal{B} = \{ \sigma \in \text{Gal}(\mathbb{Q}(X)/ \mathbb{Q}): \sqrt{p} \not \in Y, \sigma(\sqrt{p}) = \sqrt{p} \}$. Then, I define a relation $\leq$ on $\mathcal{B}$ where $\sigma \leq \tau$ if $Y_\sigma \subset Y_\tau$. This makes $(\mathcal{B},\leq)$ into a partially ordered set. If $\mathcal{C}$ is a chain in $\mathcal{B}$, then we can define a function $\sigma \in \text{Gal}(\mathbb{Q}(X)/ \mathbb{Q})$ as follows: if $x \in \mathbb{Q}(X)$, then $x \in \mathbb{Q}(\sqrt{p_1},\ldots.\sqrt{p_k})$ for some $k \in \mathbb{N}$. If $A = \{\sqrt{p_1},\ldots, \sqrt{p_k} \}$, then pick $\tau \in \mathcal{C}$ so that $Y_\tau \subset A$ is maximal (which is possible since $\mathcal{C}$ is a chain. Then, define $\sigma(x) = \tau(x)$. Since $\mathcal{C}$ is a chain, $\sigma$ is well defined and is also in $\text{Gal}(\mathbb{Q}(X)/ \mathbb{Q})$. It is also easy to see that $\sigma$ is an upper bound for $\mathcal{C}$. Therefore, every chain has an upper bound. Since $\mathcal{C}$ is nonempty (it contains the identity) it follows by Zorn's Lemma that $\mathcal{C}$ contains a maximal element. Let $\mu$ be a maximal element of $\mathcal{C}$.
I would first like to know if the above is a correct application of Zorn's Lemma. If so, then if $\tau^*$ is the maximal element, I can show that $Y_\tau = Y$.
