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mathematics, in pictures
Concept

Minimal polynomial

The smallest-degree polynomial with whole-number coefficients that a given number satisfies.

Named by 3 essays across one field — each of them below, with the objects they name alongside it.

ℚdim 12ℚ(√2)dim 22ℚ(√2, √3)dim 41√2√3√61√2√3√61√2√3√6√22√62√3√3√633√2√62√33√262 square roots taken, one at a time, and the degree doubles at each: 1 → 2 → 4the 4×4 table is the closure check — every product of basis elements landed on a whole-numbermultiple of another

Every step is a square root

A line meets a line by solving a linear equation and a circle by solving a quadratic one. There is no third case, so the numbers a construction reaches can only ever double in complexity — and a doubling is a thing that can be counted.

computation · constructible numbers x³ − 2candidatevalue thereroot?-2-10—-1-3—1-1—26—x³ − 2 has no rational root — all 4 candidates the theorem allows weretested and none is zeroa cubic with no rational root is irreducible over ℚ, so its roots have degree3

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 π = 3.141592654…coefficients from −5 to 5degreeclosest missvalue there1−x + 30.142110 tried2−2x² + 5x + 40.03121,210 tried3−x³ + 2x² + 2x + 50.016113,310 tried4−2x⁴ + 5x³ + 5x² − 4x + 30.00515146,410 tried161,040 integer polynomials of degree ≤ 4 with coefficients in [−5, 5], evaluated at π —none is zerothe same search finds x² − 2 for √2, so its silence about π is a report and not a proof

The circle that will not square

The other three impossibilities are a number having the wrong degree. This one is a number having no degree at all — and that is a claim no finite search can establish, which makes it the one place in this field where the picture has to admit what it is not doing.

computation · constructible numbers

Named alongside it

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

Constructible numberDegree of an extensionStraightedge and compassRational root theoremAlgebraic numberBasisClosureDoubling the cubeField extensionIrreducible polynomialOperation setPi

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