In general, polynomial equations higher than fourth degree are incapable of algebraic solution in terms of a finite number of additions, subtractions, multiplications, divisions, and root extractions. This was also shown by Ruffini in 1813 (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.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.Herstein, I. N. Topics in Algebra, 2nd ed. New York: Wiley, 1975.Hungerford, T. W. Algebra, 8th ed. New York: Springer-Verlag, 1997.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 Web Resource. https://mathworld.wolfram.com/AbelsImpossibilityTheorem.html