In [1], J.P. Serre proves a theorem concerning the numbers of solutions to irreducible
polynomial modulo prime numbers. In this thesis, we rst study the theorem and give
a detailed proof, and in the second part of this thesis, we study the example of the
polynomial x5⣵76;5x+12 and relate the Artin L-function of the splitting eld to modular
forms in M1(⣵76;0(1000)).
