Gauss–Wantzel theorem

The theorem characterizes constructible regular polygons: a regular polygon with n sides can be constructed using straightedge and compass exactly when n is a power of two times a product of distinct Fermat primes.

Fermat prime: A prime number of the form 2^(2^k) + 1 for a nonnegative integer k. These primes, when distinct, are precisely the odd prime factors allowed by the theorem.
Carl Friedrich Gauss: A German mathematician whose work shaped number theory, algebra, analysis, and geometry. At nineteen, he proved the regular 17-gon constructible, a landmark behind the theorem.
Regular 5-gon: A regular polygon with five equal sides and five equal angles. Five is a Fermat prime, so the theorem guarantees its straightedge-and-compass construction.
Angle trisection: The problem of dividing an arbitrary angle into three equal angles using specified geometric tools. Its general impossibility contrasts with the theorem's positive criterion for particular regular polygons.
Cyclotomic field: A field obtained by adjoining a primitive root of unity to the rational numbers. Its degree encodes the algebraic complexity of the polygon's vertices.
Disquisitiones Arithmeticae: Gauss's 1801 treatise that systematized number theory and introduced influential methods and notation. It presented the construction of the regular 17-gon and related it to arithmetic.
Regular 15-gon: A regular polygon with fifteen equal sides and fifteen equal angles. Its side count is the product of two distinct Fermat primes, 3 and 5.
Doubling the cube: The classical problem of constructing a cube with twice a given cube's volume using straightedge and compass. Its impossibility illustrates algebraic obstructions beyond the polygon criterion.
Field extension degree: The dimension of one field as a vector space over a contained field. Straightedge-and-compass constructions require relevant degrees to be powers of two.
Regular 17-gon: A regular polygon with seventeen equal sides and equal interior angles. Its construction was Gauss's celebrated example of a polygon enabled by a Fermat prime.

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