I am self studying Galois theory from Hungerford. I have questions in the proof of Fundamental theorem of Algebra(Pg 266) and got struct on proof of Fundamental Theorem of Algebra.
i have question in 2 line of 2 nd paragraph of proof . Please note that ||.5.7 is Sylow 1 st Theorem.
Kindly note that Result 3.18 there is no extension fields of dimension 2 over the field of complex numbers. ( In case it is used to deduce these). and (B) is every polynomial in $\mathbb{R}$[x] of odd degree has a root in $\mathbb{R} $
Questions: 1.How does in 2nd line of 2nd paragraph $Aut_{R} F $ must have odd index whose fixed field has odd dimension?
and in last line of page 266 how does it implies degree must be 1?
In last paragraph how does Fundamental theorem of Galois Theory implies $E_0$ is an extension of $\mathbb{C} $ with dimension $[Aut_{C} F : J]=2$ ?
I know i have asked 3 questions but I am really struck on this and would be thankful for your help.