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Concept

Radical extension

A field reached from the rationals by adjoining, one at a time, roots of numbers already present — square roots, cube roots and so on. A number is expressible by radicals exactly when it lies in such a field, and Galois theory decides which polynomials have their roots there.

Named by 3 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 Cardano's formula for x³ − 3x + 1, drawn: three real roots from conjugate cube roots. The complex plane with u³ = −0.50 + 0.866i and its conjugate, their three cube roots each, and the vertical pairings whose sums are the real roots −1.879, 0.347, 1.532.

Three real roots and no real radicals

The cubic x³ − 3x + 1 has three real roots, and Cardano's formula reaches every one of them by way of the cube roots of a complex number. That detour cannot be removed: Hölder proved in 1891 that no expression built from real radicals gives a root of an irreducible cubic whose roots are all real. The proof is three lines of the Galois correspondence, and its general form says that real radicals reach all-real roots only when square roots alone would.

algebra · Galois correspondence

Named alongside it

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

Alternating groupDiscriminantGalois groupRoots of unitySolvable groupCommutatorComplex numbersConjugacy classConstructible numberDerived seriesField extensionModular arithmetic

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