Irreducible polynomial
Named by 3 essays across one field — 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.
Named alongside it
The objects these essays reach for when they reach for this one.
Constructible numberDegree of an extensionOperation setRational root theoremStraightedge and compassCharacteristicClosureCosineCounterexampleDoubling the cubeFinite fieldMinimal polynomial