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]
https://mathshistory.st-andrews.ac.uk/Biographies/Wantzel/. Biography, 1837 construction paper and impossibility results
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Pierre Wantzel (1814–1848) was a French mathematician best known for his work in geometry and, specifically, for his contributions to the field of classical constructibility problems. He is particularly famous for proving in 1837 that certain problems, such as squaring the circle, trisecting an angle, and doubling the cube, cannot be solved using only a compass and straightedge.