Let $\mathbb{R}(t)$ be the field of rational functions over $\mathbb{R}$ (the fraction field of $\mathbb{R}[x]$).
I am looking for elements in the Brauer group of the field, and the current idea I have to follow on is to find infinitely many cyclic field extensions, and use those to create cyclic division algebras.
My Galois theory experience is not very rich with transcendental extensions of $\mathbb{R}$ and I'm a bit lost. Am I even on the right path towards the Brauer group? Any ideas on how to prove there are many cyclic extensions?