I've been stuck on two last parts for two different questions, can someone please help me with these. The first question is:
Let $\sigma\in Aut(L/\mathbb{Q})$, where $L$ is some subfield of $\mathbb{C}$, such that $\sigma:L\rightarrow L$ and $\sigma(\alpha)=\bar\alpha$. The first part of the question asks to show that $\sigma$ has order $2\iff L\not\subset \mathbb{R}$. By order does it mean, the minimum number of times I need to repeatedly call the function $\sigma$ on $\alpha$ till I arrive with output $\alpha$. If this is the case, then $\sigma(\alpha)=\alpha\neq\alpha$, however $\sigma(\sigma(\alpha))=\sigma(\bar\alpha)=\alpha$. So order is equal to 2. If I am right, then could someone help me word this correctly.
The second half of the first question is:
Let $F=L^{<\sigma>}$, show that $[L:F]=1$ or $2$, according to whether $L\subset\mathbb{R}$ or $L\not\subset\mathbb{R}$.
Where $L^{<\sigma>}:=${$\alpha\in L|\sigma_i(\alpha)=\alpha\space\space\forall\sigma\in <\sigma>$}
Is it okay for me to assume that $<\sigma>=${$\sigma, \space\sigma\circ\sigma$=$I$}, where $I$ is the identity map. Are there any elements in $<\sigma>$ that I'm missing out. Since $\sigma\circ\sigma$ will always map any $\alpha$ back to $\alpha$, so we only need to find the set of all $\alpha$, for which $\sigma(\alpha)=\alpha$.
When $L\subset\mathbb{R}\implies \sigma(\alpha)=\bar\alpha = \alpha$. It follows that $F=L$, hence $[L:F]=1$.
When $L\not\subset\mathbb{R}$, $\exists l\in L$, such that $l\in\mathbb{C}/\mathbb{R}$. This is where I'm completely stumped, can I somehow use the fact that $\sigma$ has order $2$ from the previous part, to show that $[L:K]=2$.
The Second question I'm struggling with is:
Let $\sigma (\alpha)=\bar\alpha$. I need to show that $\mathbb{Q}(\zeta)^{<\sigma>}=\mathbb{Q}(\zeta + 1/\zeta)$. That is, I need to show that $\forall\alpha\in\mathbb{Q}(\zeta + 1/\zeta)$, $\sigma_i(\alpha)=\alpha$ $\forall\sigma_i\in<\sigma>$, where $\zeta$ is the primitive $p^{th}$ root of unity and $p$ is an odd prime. From the previous questions I had proved that $\mathbb{Q}(\zeta)/\mathbb{Q}$ is a galois extension. Also, I had proved that the function $\mu:(\mathbb{Z}/p\mathbb{Z})^*\rightarrow Aut(\mathbb{Q}(\zeta)/\mathbb{Q})$, defined as $\mu(\bar a)(\zeta)=\zeta^a$. Can I use the two previous results to answer this question.