中文摘要：

目的分析Gauss（1777—1855）解方程的主要思想和具体步骤，研究其对解代数方程的贡献。方法从原始文献出发，寻求求解代数方程的内涵。结果Gauss证明了分圆方程根式可解；通过对Lagrange（1736—1813）与Gauss关于解代数方程方法的比较，得出Gauss解方程的方法是对Lagrange解方程方法的应用；最后Gauss在解方程的过程中得出一个根式扩张塔。结论Gauss不仅证明了分圆方程是根式可解的，且在解方程的过程中得到根式扩张塔，改变了方程可解的定义。

英文摘要：

Aim To investigate Gauss＇（ 1777--1855 ） main idea and procedures of solving the cyclotomic equations, and then establish Gauss＇ contribution to solving algebraic equation. Methods Search for the inner meaning of solving algebraic equations from the original literature. Results Gauss proved the algebraic solvability of cy- clotomic equation; Lagrange＇s and Gauss＇ methods of solving equation being compared, it shows that Gauss＇ method is the application of Lagrange＇s method. Gauss obtained a tower of radical field extension in solving algebraic equation. Conclusion Gauss proved the algebraic solvability of cyclotomic equations and got a tower of radical field extension, which changed the definition of radical solvability of the equation.