I'm working in the following problem from Morandi's Book Field and Galois Theory:
Let $A$ =$\left(\begin{array}{cc} a & b \\ c & d \end{array} \right)$ with $a=d=-1/2$ and $c=-b=\sqrt{3}/2$ be the matrix that rotates the plane around the origin by $120º$. Consider $\varphi_A:\mathbb{R}(x)\to \mathbb{R}(x)$ given by $\varphi_A(x)=(ax+b)/(cx+d)$ (this function lies in $Gal(\mathbb{R}(x)/\mathbb{R})$).
We have that $S=\{Id, \varphi_A,\varphi_{A^2}\}\leq Gal(\mathbb{R}(x)/\mathbb{R})$ (as one can check), consider then $F=\mathcal{F}(S)$ the fixed field of $S$. Prove that $\mathbb{R}(x)/F$ is Galois and find a $u$ so that $F=\mathbb{R}(u)$, then, find the polynomial $min(F,x)$ and find all the roots of this polynomial.
(I've omitted notations for the porpuse of brevity, I hope the statement is clear). Proving that $\mathbb{R}(x)/F$ is Galois is easy. I'm stuck in finding the $u$. I thought $u$ could be $\varphi_A(x)$, but this leads $F\subseteq \mathbb{R}(u)$ but not the converse.
Can you give me a hand with this problem please? I'm sure tips will suffice. Thanks.
It's way easier to answer the questions in reverse.
Since we know the Galois group of $\Bbb R(x)$ over $\mathcal F(S)$, the conjugates of $x$ have to be $\phi_A(x)$ and $\phi_{A^2}(x)$.
Then, the minimal polynomial of $x$ is $(T - x)(T - \phi_A(x))(T - \phi_{A^2}(x))$
You can compute everything and check that the coefficients are fixed by $S$ if you want but it is already obvious.
Finally to find such an $u$, let's pick a random element of $\mathcal F(S)$.
For example one of the coefficients of the polynomial is $u = Id(x) + \phi_A(x) + \phi_{A^2}(x) = 3x \frac {3-x^2}{1-3x^2}$.
Then $x$ is a root of $(1-3T^2)u - 3T(3-T^2)$ which is a polynomial of degree $3$, so $\Bbb R(u) \subset \Bbb R(x)$ has degree at most $3$, and so $\Bbb R(u)$ must be $\mathcal F(S)$ (it could have been a strict subfield if we were unlucky)
After dividing it by $3$ this should be the same minimal polynomial as before, but this time the coefficients are explicit rational functions of $u$.