Why is there no formula for the fifth-degree equation?

That time, that place

The formula for the quadratic had been known for ages. Cubics and quartics were cracked in sixteenth-century Italy. Then it stopped.

For three hundred years nobody found a formula for the fifth degree, and in 1824 Abel proved that none can exist.

But why none can exist — and what separates the equations that do yield from those that don't — was still open.

Why this question

Galois stopped looking at the equation and looked at the relations among its roots.

Say an equation has five roots. Certain relations among them survive if you permute the roots; some permutations preserve them, others break them.

Collect only the permutations that preserve, and the collection has a structure of its own: do two in succession and you are still inside it; undo one and you are still inside it. Galois called such a collection a group.

And he showed that whether an equation yields to a formula depends on whether this group breaks apart, step by step, in a particular way.

What was found

The group of the fifth-degree equation does not break that way. Hence no formula — not undiscovered, but impossible.

Galois died in a duel at twenty. He had been jailed twice for political activity, and the reason for the duel is still not clear. The story goes that in the margins of a letter written the night before he repeated that he had no time.

His manuscripts had earlier gone to Cauchy and to Fourier, and vanished when Fourier died soon after. Liouville put the surviving papers in order and published them fourteen years later.

The group idea travelled far past equations. It became the universal language for handling symmetry. Noether used it to draw conservation laws out of symmetries; Gell-Mann used it to arrange a hundred-odd particles.

Terms on this card

Group theory