I will talk about my recent work that establishes a correspondence between Galois extensions of rational function fields over arbitrary fields F_s and Galois extensions of the rational function field over the ultraproduct of the fields F_s. As an application, I will discuss an analogue of the Kronecker-Weber theorem for rational function fields over ultraproducts of finite fields. I will also describe an analogue of cyclotomic fields for these rational function fields that generalizes the works of Carlitz from the 1930s, and Hayes in the 1970s. If time permits, I will talk about how to use the correspondence established in my work to study the inverse Galois problem for rational function fields over finite fields.