Lior Bary-Soroker
The field crossing argument: An example
  
Abstract: 
I will present the field crossing argument via an example: 
Consider a perfect field K having a unique extension of each degree
(e.g. a finite field) 
and consider a polynomial f(T,X) in two variables with coefficients in K
that is separable and of positive degree n in X.
Assume the cyclic group C_n is the Galois group of f over L(X),
for any algebraic extension L/K.
 
We will prove that there are 
(1) a polynomial g(T,X) of X-degree n and 
(2) a finite set S
(whose size depends only on the degree of f and NOT on the field), 
such that for every a in K which is not in S the univariate polynomial  
f(a,X) is irreducible over K iff g(a,X) has a root in K.