Definition:Quintic Equation

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Let $\map f x = a x^5 + b x^4 + c x^3 + d x^2 + e x + f$ be a polynomial function over a field $\mathbb k$ of degree $5$.

Then the equation $\map f x = 0$ is the general quintic equation over $\mathbb k$.


Example: $z^5 - 2 z^4 - z^3 + 6 z - 4 = 0$

The quintic equation:

$z^5 - 2 z^4 - z^3 + 6 z - 4 = 0$

has solutions:

$1, 1, 2, -1 \pm i$

Also see

  • Abel-Ruffini Theorem, which proves that, in general, a quintic equation can not be solved analytically.

Historical Note

Ehrenfried Walther von Tschirnhaus published what he thought was a solution to the quintic equation in $1683$, but Gottfried Wilhelm von Leibniz pointed out that it was fallacious.

Leonhard Paul Euler tried and failed to solve it, but he did develop new methods for solving the quartic, as did Étienne Bézout.

Joseph Louis Lagrange consolidated everything that was known about solutions to equations of degree less than $5$ in his Réflexions sur la Résolution Algébrique des Equations of $1770$.

He noted that the usual attempt to find a solution by permutation of roots fails for the quintic.

In $1799$ Paolo Ruffini published his $2$-volume La teoria generale delle equazioni in which he tried to prove its insolubility, but he appeared to be unsuccessful.

In $1810$ he had another unsuccessful go, and again in $1813$.

Despite his lack of success, he had in fact made significant progress, despite this not having been realised at the time.

In $1824$, Niels Henrik Abel finally succeeded, although his proof was needlessly lengthy and contained a small mistake.

This was finally patched up in $1879$ by Leopold Kronecker, whose proof, while based on Niels Henrik Abel's ideas, was simple and rigorous.

The problem still remained whether certain particular quintics could be solved by radicals.

The techniques for answering this question were cracked wide open by the work of Évariste Galois.