Galois group
Named by 10 essays across 3 fields — each of them below, with the objects they name alongside it.
The lattice that runs the other way
The symmetries of a polynomial's roots form a group, and the fields between the bottom and the top form a lattice. The two are the same picture, one of them turned over — a bigger group of symmetries fixes less, so it names a smaller field.
Three ways to pair four roots
Four roots can be split into two pairs in exactly three ways, and the three numbers r·r′ + r″·r‴ those pairings give are the roots of a cubic whose coefficients can be read straight off the quartic. That cubic is where Ferrari's formula gets its cube roots, and it is also a verdict: whether its roots are rational decides which of the twenty-four symmetries the quartic's roots actually have.
How a polynomial breaks modulo the primes
Reduce x³ − 2 modulo a prime and it factors: into three linear pieces for some primes, one linear and one quadratic for others, not at all for the rest. Over the primes up to twenty thousand those three patterns occur a sixth, a half and a third of the time — exactly the shares of the identity, the flips and the rotations in the symmetry group of a triangle, the group that permutes the three cube roots of 2. A polynomial's factorisations modulo primes are a census of its Galois group.
The sums that obey a smaller equation
The twelve non-trivial thirteenth roots of unity satisfy an equation of degree twelve. Split them into three groups of four — the right three groups — and add each group: the three sums are the roots of x³ + x² − 4x + 1, an equation of degree three with whole-number coefficients. Gauss called such sums periods, found one for every divisor of p − 1, and used them to build the seventeen-gon from four quadratic equations.
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.
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.
The five crescents that can be squared
The circle cannot be squared, but some regions bounded by circular arcs can: Hippocrates of Chios found a crescent with exactly the area of a triangle around 440 BC. The trick is to make two multiples of π cancel, and it works for a crescent whose two arcs turn through angles in the ratio m to n exactly when an angle defined by a polynomial can be drawn with ruler and compass. That happens for five ratios and no others.
Twenty-seven million cubics, sorted by their symmetry
Write down every cubic x³ + ax² + bx + c with coefficients up to 150 and sort each by its Galois group. 98.7% have all six symmetries of three roots. The rest are reducible at a rate of fifteen in every H², or cyclic at five in every H to the three-halves — and the cyclic ones turn up four times as often as chance would allow, because discriminants are not random numbers.
The quartics whose roots stay paired
A quartic can have five different Galois groups, and in a box of nearly fourteen million quartics they do not come in order of size. The eight-element group D₄ outnumbers the twelve-element A₄ thirty-five to one. Every even quartic has a smaller group than S₄, and moving one step off that plane restores the full group almost everywhere.
The product raised to the twenty-fourth power
Euler's product (1 − q)(1 − q²)(1 − q³)… has the partition numbers for its reciprocal. Raise it to the twenty-fourth power and multiply by q, and the coefficients that come out — 1, −24, 252, −1472, 4830, −6048 — multiply like the primes they are indexed by, stay inside a bound Ramanujan guessed and Deligne proved fifty-eight years later, agree with a divisor sum modulo 691, and know, modulo 23, how a cubic equation factors.
Named alongside it
The objects these essays reach for when they reach for this one.
DiscriminantField extensionRoots of unityConstructible numberIrreducible polynomialAlternating groupCongruenceExhaustive searchModular arithmeticPermutationPolynomialRadical extension