Page "Polynomial" Paragraph 36

There is a difference between approximating roots and finding exact expressions for roots.

Formulas for expressing the roots of polynomials of degree 2 in terms of square roots have been known since ancient times ( see quadratic equation ), and for polynomials of degree 3 or 4 similar formulas ( using cube roots in addition to square roots ) were found in the 16th century ( see cubic function and quartic function for the formulas and Niccolo Fontana Tartaglia, Lodovico Ferrari, Gerolamo Cardano, and Vieta for historical details ).

But formulas for degree 5 eluded researchers.

In 1824, Niels Henrik Abel proved the striking result that there can be no general ( finite ) formula, involving only arithmetic operations and radicals, that expresses the roots of a polynomial of degree 5 or greater in terms of its coefficients ( see Abel-Ruffini theorem ).

In 1830, Évariste Galois, studying the permutations of the roots of a polynomial, extended Abel-Ruffini theorem by showing that, given a polynomial equation, one may decide if it is solvable by radicals, and, if it is, solve it.

This result marked the start of Galois theory and Group theory, two important branches of modern mathematics.

Galois himself noted that the computations implied by his method were impracticable.

Nevertheless formulas for solvable equations of degrees 5 and 6 have been published ( see quintic function and sextic equation ).