#math #geometry omg, reading this with ease and clarity. even 10 years ago, to understand why angle trisection is impossible require reading graduate math textbook that costs $60 for months.
In geometry and algebra, 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 in a finite number of steps. Not all real numbers are constructible and to describe those that are, algebraic techniques are usually employed. However, in order to employ those techniques, it is useful to first associate points with constructible numbers.
A point in the Euclidean plane is a constructible point if it is either endpoint of the given unit segment, or the point of intersection of two lines determined by previously obtained constructible points, or the intersection of such a line and a circle having a previously obtained constructible point as a center passing through another constructible point, or the intersection of two such circles. Now, by introducing cartesian coordinates so that one endpoint of the given unit segment is the origin (0, 0) and the other at (1, 0), it can be shown that the coordinates of the constructible points are constructible numbers.
In algebraic terms, a number is constructible if and only if it can be obtained using the four basic arithmetic operations and the extraction of square roots, but of no higher-order roots, from constructible numbers, which always include 0 and 1. 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 roots. 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.
the key is this: The set of constructible numbers are the quadratic closure of the rational numbers: the smallest field extension that is closed under square roots.