unsolvability of the quintic
For quadratics, cubics and quartics, mathematicians eventually found master formulas that produce the roots from the coefficients using only arithmetic and root extractions. For three centuries people hunted for the analogous formula for degree five. The startling answer is that no such formula can exist — not because it is hard to find, but because it is impossible.
Precisely, the Abel–Ruffini theorem states that there is no general formula in radicals that solves every quintic (degree-five) equation using only the coefficients, the four operations, and root extractions. The same impossibility holds for every degree five and higher. Galois theory pinpoints the reason: the Galois group of the general quintic is the symmetric group on 5 elements, and that group is not solvable.
Two honest cautions. First, this does not say quintics have no roots — by the fundamental theorem of algebra they always do, and numerical methods find them. Second, it does not say no particular quintic is solvable: special ones like x^5 - 2 = 0 are. The theorem rules out a single universal radical formula that works for all of them.
The quintic x^5 - x - 1 = 0 has Galois group equal to the symmetric group on 5 letters, which is not solvable; hence its roots cannot be written with any finite combination of arithmetic and radicals.
A specific unsolvable-by-radicals quintic.
Niels Henrik Abel gave a complete proof of the impossibility in 1824; Paolo Ruffini had argued it earlier with a gap. Galois's group-theoretic framework then explained exactly why, and which higher equations are or are not solvable.