Theorem of Wantzel

Abstract

In 1796, Gauss succeeded in solving the problem of constructing the regular 17-gon with compass and straightedge. Later he proved that, using a compass and straightedge, it is possible to construct the regular polygons with $n=2^m n_1\cdots n_l$ sides if $n_1,\cdots, n_l$ are different prime numbers of the form $n_k=2^{2^{ν_k}}+1$. Gauss also knew that only these regular polygons can be constructed but did not prove it.\linebreak P. Wantzel completed the result of Gauss and proved it in 1837. The present paper provides a new proof for Wantzel's theorem.