Szego Seminar: "Parametric polynomials for small Galois groups"
Abstract: Given a finite field extension, classical Galois theory allows us to associate it with a finite group. The inverse Galois problem investigates the reverse: given a finite group, can we find a finite field extension that is attached to it? If a finite field extension is a Galois extension, we call its associated finite group a Galois group. People who work on the inverse Galois problem want to know if every finite group is a Galois group. A usual way to attain a Galois extension is by taking the splitting field of some polynomial over the rational field Q. In this case, its Galois group can be viewed as a permutation group of the roots of that polynomial. This gives us some hope of studying the inverse Galois problem constructively. That is, for a given finite group G, we try to write down a polynomial over Q such that the finite group associated to its splitting field over Q is isomorphic to G. In this talk, we’ll see some families of polynomials that describe all Galois extensions with Galois group isomorphic to some certain small group.
Host: Nathan Wagner