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Concept

Field extension

A field containing a smaller one, so that the larger one's elements can be built from the smaller one's.

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

01−12√3⁄2−√3⁄2two points, 1 line and 2 circles, 4 new pointseach new point was checked to lie on two of the objects drawn before it

What two points can build

A compass and a straightedge are not a craft. They are two operations on a set of points, applied over and over, and writing them that way turns "can this be drawn?" into a question with an answer.

computation · constructible numbers 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

Named alongside it

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

ClosureConstructible numberStraightedge and compassBasisCircleDegree of an extensionGeometric meanIncidenceMinimal polynomialOperation setQuadratic extensionRational root theorem

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