Number Theory Seminar

Date: September 28, 2016

Time: 1:45PM - 2:45PM

Location: BLOC 220

Speaker: Andrew Bridy, Texas A&M University

Title: Ramification and Galois Theory of Preimage Fields

Abstract: Let K be a number field or a function field of characteristic 0, and let f in K(x) and b in K. In the tower of field extensions given by K_n := K(f^{-n}(b)), I show that the conclusion of Zsigmondy's theorem holds for ramified primes (for number fields, the proof is conditional on the abc conjecture). This has applications to the Galois theory of these extensions. In some cases, it can be used to prove a dynamical analogue of Serre's open image theorem: the projective limit of the groups Gal(K_n/K) has finite index in the automorphism group of an infinite rooted tree.

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