Arboreal Galois Representations

Abstract

We provide some general tools that can be used for polynomials in any degree to show G1 = Aut(T1). We introduce the idea of Newton irreducibility to help push us closer to a proof to Odoni's conjecture for monic integer polynomials when d = 4. We also show that current techniques used in the literature will not work in proving Odoni's conjecture for monic quartic polynomials. Finally, we look at how certain behaviors of the critical points of a polynomial f(x) 2 K[x] force G1 to have in nite index in Aut(T1).