Abel's impossibility theorem states that the general polynomial equation of degree 
cannot be solved in terms of a finite number of additions,
subtractions, multiplications,
divisions, and root extractions
(Khovanskii 2004, p. 682). This excludes a radical
formula valid for arbitrary coefficients, rather
than radical solutions of every particular polynomial
equation. Particular quintic equations and
sextic equations can be solved by radicals
(Dummit 1991, p. 387; Hagedorn 2000). Ruffini also addressed the impossibility
of a general solution (Wells 1986, p. 59).
Abel's Impossibility Theorem
See also
Cubic Equation, Galois's Theorem, Polynomial, Quadratic Equation, Quartic Equation, Quintic EquationExplore with Wolfram|Alpha
References
Abel, N. H. "Beweis der Unmöglichkeit, algebraische Gleichungen von höheren Graden als dem vierten allgemein aufzulösen." J. reine angew. Math. 1, 65, 1826. Reprinted in Abel, N. H. Œ (Ed. L. Sylow and S. Lie). Christiania [Oslo], Norway, 1881. Reprinted in New York: Johnson Reprint Corp., pp. 66-87, 1988.Artin, E. Galois Theory, 2nd ed. Notre Dame, IN: Edwards Brothers, 1944.Dummit, D. S. "Solving Solvable Quintics." Math. Comput. 57, 387-401, 1991. https://doi.org/10.1090/S0025-5718-1991-1079014-X.Faucette, W. M. "A Geometric Interpretation of the Solution of the General Quartic Polynomial." Amer. Math. Monthly 103, 51-57, 1996.Fraleigh, J. B. A First Course in Abstract Algebra, 7th ed. Reading, MA: Addison-Wesley, 2002.Hagedorn, T. R. "General Formulas for Solving Solvable Sextic Equations." J. Algebra 233, 704-757, 2000. https://doi.org/10.1006/jabr.2000.8428.Herstein, I. N. Topics in Algebra, 2nd ed. New York: Wiley, 1975.Hungerford, T. W. Algebra, 8th ed. New York: Springer-Verlag, 1997.Khovanskii, A. G. "On Solvability and Unsolvability of Equations in Explicit Form." Russian Math. Surveys 59, 661-736, 2004. https://doi.org/10.1070/RM2004v059n04ABEH000759.Tignol, J.-P. Galois' Theory of Algebraic Equations. Harlow, England: Longman Scientific & Technical, 1988.van der Waerden, B. L. A History of Algebra: From al-Khwārizmī to Emmy Noether. New York: Springer-Verlag, pp. 85-88, 1985.Wells, D. The Penguin Dictionary of Curious and Interesting Numbers. Middlesex, England: Penguin Books, p. 59, 1986.Referenced on Wolfram|Alpha
Abel's Impossibility TheoremCite this as:
Weisstein, Eric W. "Abel's Impossibility Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/AbelsImpossibilityTheorem.html