# Pierre Wantzel

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.[1]

Pierre Laurent Wantzel
Born5 June 1814
Paris, France
Died21 May 1848 (aged 33)
Paris, France
ResidenceFrance
Known forSolving several ancient Greek geometry problems
Scientific career
FieldsMathematics

In a paper from 1837,[2] Wantzel proved that the problems of

are 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:

1. 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[3] or in a textbook.[4] 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[1] that his name started to be well-known among mathematicians.[5]

Wantzel was also the first person who proved, in 1843,[6] 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.

## References

1. ^ a b c Cajori, Florian (1918). "Pierre Laurent Wantzel". Bull. Amer. Math. Soc. 24 (7): 339–347. doi:10.1090/s0002-9904-1918-03088-7. MR 1560082.
2. ^ Wantzel, L. (1837), "Recherches sur les moyens de reconnaître si un Problème de Géométrie peut se résoudre avec la règle et le compas" [Investigations into means of knowing if a problem of geometry can be solved with a straightedge and compass], Journal de Mathématiques Pures et Appliquées (in French), 2: 366–372
3. ^ Echegaray, José (1887), "Metodo de Wantzel para conocer si un problema puede resolverse con la recta y el circulo", Revista de los Progresos de las Ciencias Exactas, Físicas y Naturales (in Spanish), 22: 1–47
4. ^ Echegaray, José (1887), Disertaciones matemáticas sobre la cuadratura del círculo: El metodo de Wantzel y la división de la circunferencia en partes iguales (PDF) (in Spanish), Imprenta de la Viuda é Hijo de D. E. Aguado, retrieved 15 May 2016
5. ^ Lützen, Jesper (2009), "Why was Wantzel overlooked for a century? The changing importance of an impossibility result", Historia Mathematica, 36 (4): 374–394, doi:10.1016/j.hm.2009.03.001
6. ^ Wantzel, M. L. (1843), "Classification des nombres incommensurables d'origine algébrique" (PDF), Nouvelles Annales de Mathématiques (in French), 2: 117–127