不可約多項式模p解之探討

Abstract

J.P. Serre 在 [1] 證明了有關不可約多項式模掉質數 $p$ 之解個數的定理。本篇論文開頭描述此定理並且給完整的證明。論文第二部分將以一個五次多項式 x^5-5x+12 為例來探討並且找出此多項式對應的 Artin L-函數與 weight 為 1 、 level 為 1000的模型式之間的關係。

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 x55x+12 and relate the Artin L-function of the splitting eld to modular forms in M1(0(1000)).