Pierre Wantzel

French mathematician- Pierre Laurent Wantzel (5 June 1814 in Paris – 21 May 1848 in Paris) was a French mathematician who proved that several ancient geometric problems were impossible to solve using only compass and straightedge.In a paper from 1837, Wantzel proved that the problems of doubling the cube, and trisecting the angleare impossible to solve if one uses only compass and straightedge.
- In the same paper he also solved the problem of determining which regular polygons are constructible: a regular polygon is constructible if and only if the number of its sides is the product of a power of two and any number of distinct Fermat primes (i.e.
- that the sufficient conditions given by Carl Friedrich Gauss are also necessary)The solution to these problems had been sought for thousands of years, particularly by the ancient Greeks.
- However, Wantzel's work was neglected by his contemporaries and essentially forgotten.
- Indeed, it was only 50 years after its publication that Wantzel's article was mentioned either in a journal article or in a textbook.
- Before that, it seems to have been mentioned only once, by Julius Petersen, in his doctoral thesis of 1871.
- It was probably due to an article published about Wantzel by Florian Cajori more than 80 years after the publication of Wantzel's article that his name started to be well-known among mathematicians.Wantzel was also the first person who proved, in 1843, that when a cubic polynomial with rational coefficients has three real roots but it is irreducible in Q[x] (the so-called casus irreducibilis), then the roots cannot be expressed from the coefficients using real radicals alone, that is, complex non-real numbers must be involved if one expresses the roots from the coefficients using radicals.
- This theorem would be rediscovered decades later by (and sometimes attributed to) Vincenzo Mollame and Otto Hölder.

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