TY - JOUR
TI - On the roots of polynomials modulo primes
DO - https://doi.org/doi:10.7282/T3DN43M6
PY - 2012
AB - We study the problem of counting the number of roots of an irreducible polynomial $f(X) in mathbb{Z}[X]$ modulo rational primes. We consider the family of polynomials $f_n(X) = X^n-X-1$, which have Galois groups isomorphic to $S_n$. The approach we take is to attach Galois representations to the counting problem and then to relate these to automorphic forms. In particular, we attempt to attach the representations to holomorphic forms on $GL_2$. We show this only works when $n leq 5$, and we present the solutions to the problem in the $n = 4$ and $5$ cases, following methods due to Serre, Crespo, and Buhler for explicitly constructing Galois representations. The solution to the $n = 5$ case is novel, requiring Hilbert modular forms. In solving the problem, we produce the first example of an icosahedral Hilbert form that is not the base change of a classical form.
KW - Mathematics
KW - Polynomials
KW - Number theory
KW - Galois theory
LA - eng
ER -