我們知道二次、三次及四次的方程式都可以用根式表示，至於五次方程式的解遲遲無法解決，這個問題持續了兩三百年，直到1832年，一位法國青年Galois在其決鬥前夕，在他的遺書中，這位偉大的青年數學家引進了群的理論，證明了：五次及五次以上的方程式，不可能有公式解。從此數學家才解除了尋找公式解的惡夢。那麼我們就針對五次方程式的一些特殊型來討論x^5+px+q。而且我們會著重在S_5的子群：A_5(Alternating group)和D_5(Dihedral group)，來探討。

We know that the solutions of quadratic, cubic and quartic can be expressed by radicals. As to the solution of the quintic which has became an issue for past two or three hundred years, it was not solved until 1832 a great young Mathematician of French named Galois who recommended group theory that proved a polynomial of degree 5 and over 5 can’t be solvable by radicals. From now on, Mathematicians relieved the nightmare of looking for the solutions of radicals. 
    Then we discussed some particular models of quintic, x^5+px+q. Also our discussion will focus on the alternating group A_5 and dihedral group D_5.

1.Abstract……………………………….p.1
2.Background…………………………….p.2
3.Main Theory…………………………..p.14
4.Quintic Polynomials…………………p.26
5.Refenerce……………………………..p.38

1.Ian Stewart Galois Theory. Chapman and Hall, 1972.
2.Thomas W. Hungerford Algebra Springer-Verlag New York Inc, 1974
3.I. N. Herstein Abstract Algebra 3rd ed., Prentice-Hall International, Inc, 1996.
4.Jean-Perre Escofier Galois Theory Springer-Verlag New York Inc, 2001
5.Michael Artin Algebra Prentice-Hall Englewood Cliffs, New Jersey, 1991.
6.Roland, G. and Yui, N. and Zagier, D., A parametric family of quintic polynomials with Galois group D_5. J. Number Theory 15 (1982), no. 1,137-142.