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For $N\ge 3$, a regular polygon with $N$ sides is a constructible regular polygon exactly when $N=2^a{p}_1\cdots p_r$, where $a$ is a nonnegative integer and the $p_j$ are distinct Fermat primes; the product may be empty. Thus seventeen and fifteen qualify, while nine does not. Carl Friedrich Gauss established sufficiency, and Pierre Wantzel proved necessity in 1837.^[ref]

Pierre Wantzel proved impossibility results for straightedge and compass constructions, including general angle trisection and doubling the cube. His 1837 work translated geometric constructions into restrictions on algebraic numbers.

Each permitted construction extends the coordinate field by at most a square root. Constructing a number of irreducible degree three over the rationals therefore cannot be achieved by such a tower of quadratic extensions. This explains why failure to find a construction can have a structural mathematical cause rather than merely reflecting lack of ingenuity.^[ref]