Absolute Galois Group of $\mathbb{R}(t)$.

Let us consider the field of rational functions in one variable with real coefficients, $\mathbb{R}(t)$. The algebraic closure is the field of algebraic functions with real coefficients. What is known of the absolute galois group here? Can we say anything, and if sp, what?

Solution 1:

Let $k$ be an algebraically closed field of cardinality of $\kappa$. Then the absolute Galois group of $k(t)$ is the free profinite group $\hat{F}_\kappa$ on $\kappa$ generators. See here.

So in your case we have that the absolute Galois group of $\mathbb R(t)$ is $\mathbb Z/ 2\mathbb Z \rtimes \hat{F}_\kappa$. Since the absolute Galois group will be generated by complex conjugation and the absolute Galois group of $\mathbb C(t)$.