The Galois theory of the lemniscate

Abstract

[EN] Geometric constructions with straightedge and compass go as far back as to the time of the ancient Greeks and Egyptians. In fact, there are three classical problems in Greek mathematics related to straightedge and compass constructions which were extremely important in the development of geometry, which are the squaring of a circle, the duplication of a cube, and the trisection of an angle. These three problems are impossible to solve with straightedge and compass, but the Greeks lacked the mathematical development needed to prove this. The proof of the impossibility of these problems had to await the mathematics of the 19-th century, in particular to the development of Galois theory. In 1837 Pierre Wantzel published a paper in which he proved that duplication of a cube and the trisection of an angle are impossible to solve with straightedge and compass. The imposibility of the squaring of the circle will have to await until 1882 when Ferdinand von Lindemann proved that π is trascendental. In his paper of 1837, Pierre Wantzel also characterized which regular polygons are constructible with straightedge and compass. Sufficiency was proved by Gauss in 1796, but Wantzel was the first to prove the characterization, in a theorem now known as the Gauss-Wantzel Theorem.