Constructible number

A real number obtainable from rational numbers by finitely many arithmetic operations and square roots. Equivalently, it can be represented as a length constructible with an unmarked straightedge and compass.

Field extension: An inclusion of fields that allows elements of a larger field to be expressed using a smaller one. Each construction step enlarges the current coordinate field by at most a quadratic extension.
Euclidean geometry: The geometry of points, lines, angles, and distances governed by Euclid's parallel postulate. Classical straightedge-and-compass constructions are formulated within this geometric setting.
Doubling the cube: The problem of constructing a cube whose volume is twice that of a given cube. Its required length involves the nonconstructible cube root of two.
Transcendental number: A number that is not a root of any nonzero polynomial with rational coefficients. No transcendental number can be constructible, since construction steps yield algebraic numbers.
Quadratic extension: A field extension whose dimension over its base field is two. Intersecting constructed lines and circles introduces solutions of linear or quadratic equations.
Rational number: A number expressible as the ratio of two integers with a nonzero denominator. Integer lengths and their ratios provide the starting quantities for constructible numbers.
Squaring the circle: The problem of constructing a square with the same area as a given circle using straightedge and compass. The required length depends on pi, which is transcendental and therefore not constructible.
Algebraic number of odd degree: An algebraic number whose minimal polynomial over the rationals has odd degree greater than one. Its degree cannot be a power of two, ruling out straightedge-and-compass construction.
Minimal polynomial: The monic polynomial of least degree over a field that has a given algebraic element as a root. Its degree over the rationals must be a power of two for a constructible number.
Field: A set with addition, subtraction, multiplication, and division by nonzero elements satisfying the field axioms. Constructible coordinates form a field built from the rational numbers.

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