A constructible number is the coordinate of a point obtainable from rational starting data by straightedge-and-compass constructions.
A finite real extension is constructible when it is contained in a towerEquivalently, every element of can be obtained from rational numbers by field operations and successive square roots.
The real cyclotomic fieldis Galois of degree eight with cyclic Galois group . A subgroup chain of successive index two gives a tower of three quadratic extensions, proving that is constructible.
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A **constructible number** is a number that can be constructed using a finite number of operations involving basic geometric tools: a compass and a straightedge. This means that the number can be represented through a series of steps including drawing straight lines, constructing circles, and finding points of intersection, starting from a point set at a distance of one unit.