Rational root theorem
Named by 4 essays across one field — each of them below, with the objects they name alongside it.
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.
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.
Which polygons can be drawn
Three sides yes, seven no, seventeen yes. The list of constructible regular polygons is neither everything nor almost nothing, and the pattern in it is a fact about which numbers are one less than a power of two.
Named alongside it
The objects these essays reach for when they reach for this one.
Constructible numberDegree of an extensionStraightedge and compassIrreducible polynomialMinimal polynomialOperation setBasisClosureCosineCyclic groupDoubling the cubeFermat prime