A sextic equation is a polynomial equation of degree 6. Dividing by the leading
coefficient gives the form
A solution using genus-2 Riemann
theta functions is described by Coble (1911b, pp. 321-323).
An irreducible sextic over 
is solvable by radicals iff
its Galois group is a solvable
group. Hagedorn (2000) gives formulas for all such sextics, organized by their
transitive Galois group as a subgroup
of the symmetric group 
. This does not give a radical
solution of the general sextic, which is excluded by Abel's
impossibility theorem.
See also
Cubic Equation,
Quadratic Equation,
Quartic Equation,
Quintic
Equation
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References
Coble, A. B. "The Reduction of the Sextic Equation to the Valentiner Form--Problem." Math. Ann. 70, 337-350, 1911a.Coble,
A. B. "An Application of Moore's Cross-Ratio Group to the Solution of the
Sextic Equation." Trans. Amer. Math. Soc. 12, 311-325, 1911b.
https://doi.org/10.1090/S0002-9947-1911-1500891-5.Cole,
F. N. "A Contribution to the Theory of the General Equation of the Sixth
Degree." Amer. J. Math. 8, 265-286, 1886.Hagedorn,
T. R. "General Formulas for Solving Solvable Sextic Equations." J.
Algebra 233, 704-757, 2000. https://doi.org/10.1006/jabr.2000.8428.Trott,
M. "Solution of Equations." §1.5 in The
Mathematica GuideBook for Symbolics. New York: Springer-Verlag, 2006. https://doi.org/10.1007/0-387-28815-5.
https://www.mathematicaguidebooks.org/.Referenced
on Wolfram|Alpha
Sextic Equation
Cite this as:
Weisstein, Eric W. "Sextic Equation."
From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/SexticEquation.html
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