Irreducible polynomial
Named by 12 essays across 4 fields — each of them below, with the objects they name alongside it.
The cube that will not double
Doubling a cube needs an edge in the ratio of the cube root of two. That number satisfies an equation of degree three, three does not divide any power of two, and the oldest open problem in geometry closes in a line.
The angle that will not divide by three
Halving an angle costs one circle. Cutting it in three means solving a cubic, and for sixty degrees that cubic has no rational root — but plenty of angles do trisect, and which ones is a question with a countable answer.
The field with four elements
The integers modulo four are not a field: two times two is zero and two has no reciprocal. There is nevertheless a field with four elements, and building it means giving up on counting as the way to make arithmetic finite.
A tower whose degrees multiply
Treat a field containing another as a vector space over it, and the size of an extension becomes a dimension — one that multiplies along a tower, so that three impossible constructions become arithmetic about which numbers divide which.
A quintic a sliding mark reaches
The eleven-sided polygon needs a number of degree five, and five is not a product of twos and threes — so no conic and no angle trisector reaches it. A ruler with two scratches does, which places the marked ruler strictly above the conics and leaves its exact reach unknown.
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.
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 sine of one degree
Every table of sines ever made by hand needed the sine of one degree, and no construction with ruler and compass can produce it: one degree is a third of three, and thirds are what the constructions cannot make. Ptolemy fenced it in with an inequality and got it right to the precision his table carried. Thirteen centuries later Jamshīd al-Kāshī wrote the triple-angle formula as a rule for improving a guess and let it run, gaining almost three decimal places at every turn — a cubic solved without Cardano, by doing the same thing again.
Two fields that every prime confuses
Reduce x⁸ − 3 and x⁸ − 48 modulo any prime other than 2 and 3 and factor them: the two lists of factor degrees are the same, at all 9,590 such primes below 100,000. The fields the two equations define are still different. The reason is a pair of ways for one group to permute eight roots, with the same cycle lengths element by element and different fixed points — the same coincidence, in degree seven, as the points and lines of the Fano plane.
![Three roots that one map rotates, for every t. Shanks cubic roots for t in [-9,6]; at t=2: 3.507019, -0.221876, -1.285142; map permutation 1,2,0.](/f/three-roots-that-one-map-rotates-for-every-t-bf803443.svg)
A line of cubics that are all cyclic
In a box of cubics the full group of six symmetries is almost universal, and the three rotations are a rare exception. Daniel Shanks's x³ − tx² − (t + 3)x − 1 has the rotations at every whole t, and at every rational one. Its discriminant is the square of t² + 3t + 9, its roots are turned into one another by x ↦ −1/(1 + x), and on a suitable circle every member is an equilateral triangle.
Differences round a ring
Write four numbers at the corners of a square and put on each side the difference of its two corners. Repeat with the new square. Whatever the four whole numbers are, the process reaches nought within a few steps. Write five numbers and it almost never does. Rings whose size is a power of two always finish, and every other ring settles into a cycle whose length depends on how a polynomial factors over two elements: 7 at seven, 341 at eleven, 31 at thirty-one.
Named alongside it
The objects these essays reach for when they reach for this one.
Constructible numberGalois groupDegree of an extensionField extensionFinite fieldDiscriminantExhaustive searchOperation setCubicCyclic groupIterationMinimal polynomial