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Certain convariants of the quintic involve the same determinant factors as appeared in the system of the quartic; these are f, H, i, T and j, and are of special importance.
Hermite expresses the quintic in a forme-type in which the constants are invariants and the variables linear covariants.
Sylvester showed that the quintic might, in general, be expressed as the sum of three fifth powers, viz.
His first notable work was a proof of the impossibility of solving the quintic equation by radicals.