Concept

Solvable group

A finite group that can be broken down, one normal subgroup at a time, into pieces that are all abelian. A polynomial can be solved by radicals exactly when its Galois group is solvable, which is why the general quintic cannot be.

Named by 2 essays across one field — each of them below, with the objects they name alongside it.

The derived series of S3, S4, S5. A table with one row per group giving the sizes along its derived series, each step the subgroup generated by all commutators of the last, and whether the series reaches the identity.

The group that will not come apart

Solving an equation by radicals means building a tower of roots, and a tower of roots corresponds to a chain of subgroups with abelian steps. For the general equation of degree five that chain would have to descend through a group of sixty elements with no normal subgroup in it — so there is no formula, and the obstruction is a finite object that can be written out.

algebra · Galois correspondence

The twenty maps x ↦ ax + b modulo 5, as permutations of five roots. A four-by-five grid of pentagons; in each, arrows show where the map ax + b modulo 5 sends each of the five corners. Rows are the slopes 1, 4, 2, 3; columns the shifts 0 to 4.

The quintics that have a formula

No formula solves every equation of degree five, and yet x⁵ − 2 is solved by a fifth root and x⁵ − 5x + 12 by a longer expression of the same kind. A quintic can be solved by radicals exactly when the symmetries of its roots fit inside one group of twenty — the maps x ↦ ax + b on the numbers modulo 5 — and two exact tests on its coefficients say whether they do.

algebra · Galois correspondence

Named alongside it

The objects these essays reach for when they reach for this one.

Alternating group Radical extension Commutator Conjugacy class Derived series Discriminant Galois group Modular arithmetic Normal subgroup Permutation Roots of unity Simple group

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