Number Theory Working Seminar
Abel Castillo
UIC
Specializations of polynomials over global fields and their Galois groups
Abstract: Let $K$ be a global field, $\mathcal O_K$ its ring of integers, and $t=(t_1, t_2, \cdots, t_s)$ a set of $s$ parameters.
For $f_t \in \mathcal O_K[t,x]$, let $f_\theta$ be the specialization of $f_t$ at $\theta \in \mathcal O_K^s$.
In this talk we give a careful description of when one can embed $Gal(f_\theta / K)$ in $Gal(f_t / K(t))$, and of additional criteria for determining their equality.
Our technique will make use of basic facts from algebraic number theory over Dedekind domains, as well as a theorem of Emmy Noether regarding the existence of forms that can "detect"
when a polynomial in several variables is absolutely irreducible over a particular field.
Wednesday April 3, 2013 at 9:30 AM in SEO 512