Constructible number

A point in the Euclidean plane is a constructible point if, given a fixed coordinate system (or a fixed line segment of unit length), the point can be constructed with unruled straightedge and compass. A complex number is a constructible number if its corresponding point in the Euclidean plane is constructible from the usual x- and y-coordinate axes.

It can then be shown that a real number r is constructible if and only if, given a line segment of unit length, a line segment of length |r| can be constructed with compass and straightedge. It can also be shown that a complex number is constructible if and only if its real and imaginary parts are constructible.

In terms of algebra, a number is constructible if and only if it can be written using the four basic arithmetic operations and the extraction of square roots but of no higher-order roots. The set of constructible numbers can be completely characterized in the language of field theory: the constructible numbers form the quadratic closure of the rational numbers: the smallest field extension that is closed under square root and complex conjugation. This has the effect of transforming geometric questions about compass and straightedge constructions into algebra. This transformation leads to the solutions of many famous mathematical problems, which defied centuries of attack.

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