Number Theory Working Seminar
Abel Castillo
UIC
Probabilistic Galois Theory over number fields
Abstract: Let $k$ be a number field and $\mathcal O_k$ its ring of integers. Given a family of polynomials $f_t(X)$ in $\mathcal O_k[X,t]$, one expects that the Galois group of $f_\alpha(X)$ over $k$ (viewed as the group of permutations of its roots) is isomorphic to the generic Galois group of $f_t(X)$ over $k(t)$ for almost all $\alpha$ in $\mathcal O_k$. We study a quantitative result of S.D. Cohen, where one compares these Galois groups for $s$-parameter families of polynomials. In particular, Cohen applies a large sieve inequality to obtain a square-root saving on the number of "exceptional" elements in $\mathcal O_k^s$ of bounded height. Our goal is to understand the proof of this result and generalizations to other arithmetic contexts.
Wednesday March 6, 2013 at 9:30 AM in SEO 512