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Concept

Irreducible polynomial

A polynomial that cannot be written as a product of two smaller ones with coefficients from the same field. It is what plays the role of a prime in a polynomial ring, and quotienting by one is how a field extension is built.

Named by 6 essays across 2 fields — each of them below, with the objects they name alongside it.

Every rational number that could be a root of x³ − 2. A table of the candidate rational roots allowed by the rational root theorem, with the polynomial's exact value at each.

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.

computation · Constructible numbers Which angles with a rational cosine can be cut in three. A dial of angles marked trisectable or not, beside the cubic whose rational roots decided each one.

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.

computation · Constructible numbers The arithmetic of GF(4), and of the integers mod 4. Addition and multiplication tables of a finite field, optionally beside the table of a ring of the same kind of size.

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.

computation · Finite fields The tower ℚ ⊂ ℚ(√2) ⊂ ℚ(√2, √3). A tower of field extensions with the degree of each step, beside the multiplication table of the basis.

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.

algebra · Field extensions The polynomial the 11-sided polygon needs. The minimal polynomial of twice the cosine of the central angle of an 11-sided polygon, with its degree, the rational root test applied to it, and whether that degree is reachable by cubic steps.

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.

computation · Neusis How often each factor pattern occurs, against the Galois group. Bars for four polynomials — x³ − 3x + 1, x³ − 2, x⁴ − 10x² + 1, x⁴ − 2 — giving the share of primes up to 20000 with each factorisation pattern, beside the predicted share from each Galois group.

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.

algebra · Field extensions

Named alongside it

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

Constructible numberDegree of an extensionField extensionOperation setFinite fieldMinimal polynomialModular arithmeticRational root theoremStraightedge and compassBasisCharacteristicClosure

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