A general quintic cannot be solved algebraically in terms of finite additions, multiplications, and root extractions, as rigorously demonstrated by Abel and Galois.
Euler reduced the general quintic to
(1) 
(2) 
(3) 
Consider the quintic
(4) 
(5) 
(6) 
(7)  
(8)  
(9)  
(10)  
(11)  
(12) 
(13)  
(14)  
(15)  
(16) 
Spearman and Williams (1994) show that an irreducible quintic
(17) 
(18)  
(19) 
(20) 
(21)  
(22)  
(23)  
(24)  
(25)  
(26)  
(27)  
(28)  
(29) 
(30) 
(31) 
(32)  
(33)  
(34)  
(35) 
(36) 
(37) 
(38)  
(39)  
(40)  
(41)  
(42)  

(43) 
Felix Klein used a Tschirnhausen Transformation to reduce the general quintic to the form
(44) 

(45) 
(46) 
(47) 
(48)  
(49)  
(50)  
(51) 
(52)  
(53)  
(54)  
(55)  
(56) 
(57) 
Cadenhad, Young, and Runge showed in 1885 that all irreducible solvable quintics with Coefficients of
, , and missing have the following form
(58) 
See also Bring Quintic Form, BringJerrard Quintic Form, Cubic Equation, de Moivre's Quintic, Principal Quintic Form, Quadratic Equation, Quartic Equation, Sextic Equation
References
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Chowla, S. ``On Quintic Equations Soluble by Radicals.'' Math. Student 13, 84, 1945.
Cockle, J. ``Sketch of a Theory of Transcendental Roots.'' Phil. Mag. 20, 145148, 1860.
Cockle, J. `` On Transcendental and Algebraic SolutionSupplemental Paper.'' Phil. Mag. 13, 135139, 1862.
Davis, H. T. Introduction to Nonlinear Differential and Integral Equations. New York: Dover, p. 172, 1960.
Dummit, D. S. ``Solving Solvable Quintics.'' Math. Comput. 57, 387401, 1991.
Glashan, J. C. ``Notes on the Quintic.'' Amer. J. Math. 8, 178179, 1885.
Harley, R. ``On the Solution of the Transcendental Solution of Algebraic Equations.'' Quart. J. Pure Appl. Math. 5, 337361, 1862.
Hermite, C. ``Sulla risoluzione delle equazioni del quinto grado.'' Annali di math. pura ed appl. 1, 256259, 1858.
King, R. B. Beyond the Quartic Equation. 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, 823825, 1991.
Rosen, M. I. ``Niels Hendrik Abel and Equations of the Fifth Degree.'' Amer. Math. Monthly 102, 495505, 1995.
Shurman, J. Geometry of the Quintic. New York: Wiley, 1997.
Spearman, B. K. and Williams, K. S. ``Characterization of Solvable Quintics .'' Amer. Math. Monthly 101, 986992, 1994.
Young, G. P. ``Solution of Solvable Irreducible Quintic Equations, Without the Aid of a Resolvent Sextic.'' Amer. J. Math. 7, 170177, 1885.
© 19969 Eric W. Weisstein