Birkhoff, G. and Mac Lane, S. "Insolvability of Quintic Equations." §15.8 in 現代代數學概覽，第5版 New York: Macmillan, pp. 418-421, 1996.Chowla, S. "On Quintic Equations Soluble by Radicals." Math. Student13, 84, 1945.Cockle, J. "Sketch of a Theory of Transcendental Roots." Phil. Mag.20, 145-148, 1860.Cockle, J. "On Transcendental and Algebraic Solution--Supplemental Paper." Phil. Mag.13, 135-139, 1862.Davis, H. T. 非線性微分和積分方程導論。 New York: Dover, p. 172, 1960.Drociuk, R. J. "On the Complete Solution to the Most General Fifth Degree Polynomial." 3 May 2000. http://arxiv.org/abs/math.GM/0005026.Dummit, D. S. "Solving Solvable Quintics." Math. Comput.57, 387-401, 1991.Glashan, J. C. "Notes on the Quintic." Amer. J. Math.8, 178-179, 1885.Green, M. L. "On the Analytic Solution of the Equation of Fifth Degree." Compos. Math.37, 233-241, 1978.Harley, R. "On the Solution of the Transcendental Solution of Algebraic Equations." Quart. J. Pure Appl. Math.5, 337-361, 1862.Harley, R. "A Contribution to the History of the Problem of the Reduction of the General Equation of the Fifth Degree to a Trinomial Form." Quart. J. Math.6, 38-47, 1864.Hermite, C. "Sulla risoluzione delle equazioni del quinto grado." Annali di math. pura ed appl.1, 256-259, 1858.King, R. B. 超越四次方程。 Boston, MA: Birkhäuser, 1996.King, R. B. and Cranfield, E. R. "An Algorithm for Calculating the Roots of a General Quintic Equation from Its Coefficients." J. Math. Phys.32, 823-825, 1991.Klein, F. "Sull' equazioni dell' Icosaedro nella risoluzione delle equazioni del quinto grado [per funzioni ellittiche]." Reale Istituto Lombardo, Rendiconto, Ser. 210, 1877.Klein, F. "Über die Transformation der elliptischen Funktionen und die Auflösung der Gleichungen fünften Grades." Math. Ann.14, 111-144, 1879.Klein, F. 二十面體與五次方程解法講義。 New York: Dover, 1956.Livio, M. 無法解出的方程。 New York: Simon & Schuster, 2006.Pierpont, J. "Zur Entwicklung der Gleichung V. Grades (bis 1858)." Monatsh. für Math. und Physik6, 15-68, 1895.Rosen, M. I. "Niels Hendrik Abel and Equations of the Fifth Degree." Amer. Math. Monthly102, 495-505, 1995.Runge, C. "Ueber die aufloesbaren Gleichungen von der Form ." Acta Math.7, 173-186, 1885.Shurman, J. 五次方程的幾何。 New York: Wiley, 1997.Spearman, B. K. and Williams, K. S. "Characterization of Solvable Quintics ." Amer. Math. Monthly101, 986-992, 1994.Trott, M. "Solution of Quintics with Hypergeometric Functions." §3.13 in Mathematica 符號計算指南。 New York:Springer-Verlag, pp. 1110-1124, 2006. http://www.mathematicaguidebooks.org/. Trott, M. and Adamchik, V. "Solving the Quintic with Mathematica." http://library.wolfram.com/infocenter/TechNotes/158/. . "Solving the Quintic." Poster. Champaign, IL: , 1995. https://store.wolfram.com/view/misc/popup/solving-tqp.html.Young, G. P. "Solution of Solvable Irreducible Quintic Equations, Without the Aid of a Resolvent Sextic." Amer. J. Math.7, 170-177, 1885.
、亞迴圈群 
、二面體群 
、交錯群 
或迴圈群 
，如上所示。五次方程的可解性取決於其對應的群是否為可解群。一個具有可解迴圈群的五次方程的例子是
的計算中。
![tan[1/4sin^(-1)((16)/(25sqrt(5)rho^2))]](/images/equations/QuinticEquation/Inline9.svg)
![{-sgn(I[rho]) for R[rho]=0; sgn(R[rho]) for R[rho]!=0](/images/equations/QuinticEquation/Inline12.svg)
是橢圓模量，則原始五次方程的根由下式給出
![(-1)^(3/4)b{[m(e^(-2pii/5)q^(1/5))]^(1/8)+i[m(e^(2pii/5)q^(1/5))]^(1/8)}{[m(e^(-4pii/5)q^(1/5))]^(1/8)+[m(e^(4pii/5)q^(1/5))]^(1/8)}{[m(q^(1/5))]^(1/8)+q^(5/8)(q^5)^(-1/8)[m(q^5)]^(1/8)}](/images/equations/QuinticEquation/Inline19.svg)
![b{-[m(q^(1/5))]^(1/8)+e^(3pii/4)[m(e^(2pii/5)q^(1/5))]^(1/8)}×{e^(-3pii/4)[m(e^(-2pii/5)q^(1/5))]^(1/8)+i[m(e^(4pii/5)q^(1/5))]^(1/8)}{i[m(e^(-4pii/5)q^(1/5))]^(1/8)+q^(5/8)(q^5)^(-1/8)[m(q^5)]^(1/8)}](/images/equations/QuinticEquation/Inline22.svg)
![b{e^(-3pii/4)[m(e^(-2pii/5)q^(1/5))]^(1/8)-i[m(e^(-4pii/5)q^(1/5)]^(1/8))}×{-[m(q^(1/5))]^(1/8)-i[m(e^(4pii/5)q^(1/5))]^(1/8)}×{e^(3pii/4)[m(e^(2pii/5)q^(1/5))]^(1/8)+q^(5/8)(q^5)^(-1/8)[m(q^5)]^(1/8)}](/images/equations/QuinticEquation/Inline25.svg)
![b{[m(q^(1/5))]^(1/8)-i[m(e^(-4pii/5)q^(1/5))]^(1/8)}×{-e^(3pii/4)[m(e^(2pii/5)q^(1/5))]^(1/8)-i[m(e^(4pii/5)q^(1/5))]^(1/8)}{e^(-3pii/4)[m(e^(-2pii/5)q^(1/5))]^(1/8)+q^(5/8)(q^5)^(-1/8)[m(q^5)]^(1/8)}](/images/equations/QuinticEquation/Inline28.svg)
![b{[m(q^(1/5))]^(1/8)-e^(-3pii/4)[m(e^(-2pii/5)q^(1/5))]^(1/8)}×{-e^(3pii/4)[m(e^(2pii/5)q^(1/5))]^(1/8)+i[m(e^(-4pii/5)q^(1/5))]^(1/8)}{-i[m(e^(4pii/5)q^(1/5))]^(1/8)+q^(5/8)(q^5)^(-1/8)[m(q^5)]^(1/8)}.](/images/equations/QuinticEquation/Inline31.svg)
、
和 
的不可約可解五次方程具有以下形式
和 
是有理數。
、
和 
使得
![I(z,1,Z)=z^5(-1+11z^5+z^(10))^5
-[1+z^(30)-10005(z^(10)+z^(20))+522(-z^5+z^(25))]^2Z=0,](/images/equations/QuinticEquation/NumberedEquation11.svg)
是 
、
和 
的根式函式。該方程的解可以用超幾何函式表示為
![product_(j=0)^4[x-(omega^ju_1+omega^(4j)u_2)]=0,](/images/equations/QuinticEquation/NumberedEquation14.svg)
，
和 
是複數，這與棣莫弗五次方程 (Spearman 和 Williams 1994) 相關，並將其推廣到![product_(j=0)^4[x-(omega^ju_1+omega^(2j)u_2+omega^(3j)u_3+omega^(4j)u_4)]=0.](/images/equations/QuinticEquation/NumberedEquation15.svg)
![(omega^ju_1+omega^(2j)u_2+omega^(3j)u_3+omega^(4j)u_4)^5-5U(omega^ju_1+omega^(2j)u_2+omega^(3j)u_3+omega^(4j)u_4)^3-5V(omega^ju_1+omega^(2j)u_2+omega^(3j)u_3+omega^(4j)u_4)^2+5W(omega^ju_1+omega^(2j)u_2+omega^(3j)u_3+omega^(4j)u_4)+[5(X-Y)-Z]=0,](/images/equations/QuinticEquation/NumberedEquation16.svg)
s 滿足
![5[(u_1^3u_3u_4+u_2^3u_1u_3+u_3^3u_3u_4+u_4^3u_1u_2)-(u_1u_3^2u_4^2+u_2u_1^2u_3^2+u_3u_2^2u_4^2+u_4u_1^2u_2^2)]-(u_1^5+u_2^5+u_3^5+u_4^5)=b](/images/equations/QuinticEquation/Inline131.svg)
." Acta Math. 7, 173-186, 1885.Shurman, J. 
." Amer. Math. Monthly 101, 986-992, 1994.Trott, M. "Solution of Quintics with Hypergeometric Functions." §3.13 in 
Trott, M. and Adamchik, V. "Solving the Quintic with Mathematica."