Discriminant
Named by 15 essays across 4 fields — each of them below, with the objects they name alongside it.
What the coefficients already know
Finding the roots of a polynomial is hard and often impossible in closed form. Reading off their sum, their product and how many of them are real is none of those things — those numbers are sitting in the coefficients, and no root-finding is required to get at them.
A shared root, found without finding it
Two polynomials have a root in common exactly when one determinant built from their coefficients is zero. No root is computed, nothing is approximated, and the same construction turns two equations in two unknowns into one equation in one.
The square that cannot be negative
Cauchy–Schwarz is the load-bearing inequality of the whole subject and is nearly always stated without proof. It is one line away from a fact nobody would argue with, and the line is a parabola with no room to cross the axis.
One sign decides which curve
The general quadratic in two variables has six coefficients and draws a conic. Which of the four it draws is settled by a single combination of three of them, and the other three cannot change the answer however they are chosen.
Six points on a conic, and the line they share
Put six points on an ellipse, join them into a hexagon, and extend each pair of opposite sides until they meet. The three meeting points always lie on one straight line. The statement uses no length, no angle and no focus — which is why it holds for every conic at once, and why a straightedge alone can draw the curve through any five points.
Where two roots run into each other
Put every quadratic equation at a point of a plane, one coordinate per coefficient. Each possible root becomes a straight line there, every one of those lines touches the same parabola, and that parabola is the discriminant — the crease where the plane of roots is folded onto the plane of equations.
A quantifier is a shadow
'There is an x such that …' asks whether a column of a grid contains a mark — which is the same as asking whether a shape casts a shadow on the axis below it. Over the real numbers every such shadow can be described without the quantifier, by polynomial inequalities: 'x² + ax + 1 = 0 has a solution' is just a² ≥ 4. Over the whole numbers the same kind of shadow can carve out the primes, and any set a computer can list.
Factoring uniquely with no way to divide
Unique factorisation is proved by dividing with a small remainder, and in the Gaussian integers that works because discs of radius one cover the plane. In the integers of ℚ(√−19) the discs leave holes, no division algorithm of any kind can be made to work — and factorisation is unique anyway. The same field is why n² + n + 41 is prime forty times running.
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.
Counting the classes that break factorisation
The class number measures how badly unique factorisation fails in a field, and defined through ideals it looks impossible to compute. Gauss computed it by hand, for every field he wanted, by counting quadratic forms — and every form can be squeezed, by changes of variable that keep its values, into exactly one small standard shape.
Euler's sixty-five convenient numbers
For some n, whether a prime can be written as x² + ny² is settled by its remainder on division by 4n alone, the way a prime's remainder on division by four settles whether it is a sum of two squares. Euler found sixty-five such n, from 1 to 1,848, called them convenient, and used the largest to prove that 18,518,809 is prime. Every one of them is a number whose class group has no element of order more than two — and whether the list is complete is still not known.
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.
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.
Named alongside it
The objects these essays reach for when they reach for this one.
Galois groupPolynomialSymmetric functionProjectionQuadratic formAlternating groupClass groupClass numberCoefficientComplex numbersConicEllipse