Pierre Wantzel

Pierre-Laurent Wantzel (born June 5, 1814 in Paris ; † May 21, 1848 there ) was a French mathematician . He is known for solving two of the classic problems of ancient mathematics that have been open for centuries: proofs of impossibility for the general case of triangular division and cube doubling .

Life

Wantzel's parents were Frédéric Wantzel, professor of applied mathematics at the École speciale du Commerce in Paris, and Marie b. Aldon-Beaulieu. He spent his childhood in Écouen near Paris.

In 1826 Wantzel entered the École des Arts et Métiers de Châlons , where he was taught mathematics by Étienne Bobillier . In 1828 he moved to the Collège Charlemagne . He was a lively and brilliant student who received both first prize for a French essay at the Collège Charlemagne in 1831 and for a Latin essay in a general competition. He was in 1832 first in the entrance exams of the École polytechnique (and first in the examinations in the natural sciences for admission to the École normal supérieure ) and studied from 1832 at the École Polytechnique. In 1834 he continued his studies at the École des ponts et chaussées , where he was trained as an engineer. He was then a tutor and from 1843 examiner for analysis at the École polytechnique and he was professor for applied mechanics at the École des ponts et chaussées from 1841 .

On February 21, 1842 Wantzel married the daughter of his former teacher of Greek and Latin at the Collège Charlemagne (with the name Lievyns) and had two daughters.

His interests were wide-ranging: he immersed himself in Scottish and German philosophy, studied history and music in addition to mathematics and took part in debates. Before his marriage, he had an erratic lifestyle that ultimately ruined his health. He devoted himself entirely to his studies, hardly slept, ate only irregularly and kept himself awake with coffee and opium, as Adhémar Jean Claude Barré de Saint-Venant lamented in his obituary after his untimely death.

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As a schoolboy at the age of 15 (1829), Wantzel published the evidence of a widespread, but hitherto unproven method for determining square roots for the new edition of a school textbook by Antoine André Louis Reynaud .

Wantzel showed in a work from 1837 that there can be no construction with compasses and ruler for doubling the cube and for dividing angles into three . Both boil down to proving that, in general, it is not possible to represent cubic irrational (solutions to cubic equations) by expressions with square roots. This is often shown today with Galois theory (after the unpublished work of Évariste Galois, who also died young in 1832 ) and then amounts to the fact that a cubic extension of the field of rational numbers has a degree of transcendence that can be divided by three, while numbers that can be constructed with compasses and ruler belong to an expansion field with the degree of a power of two and must therefore be different. Since this is based on elementary questions of divisibility for natural numbers, the full group-theoretical "machinery" of Galois theory is unnecessary in this case.

He also showed that the number of sides of a constructible polygon must meet the numerical condition of being a product of a power of two and different Fermat prime numbers (if this condition is met, as Carl Friedrich Gauß had already shown in his Disquisitiones Arithmeticae) construction is possible).

More precisely, he showed that numbers that can be constructed with compasses and ruler must satisfy an irreducible polynomial equation of degree ( ), then he showed that ${\displaystyle 2^{k}}$ ${\displaystyle k\geq 1}$

• in the case of cube doubling, the relevant equation is irreducible, and that ${\displaystyle x^{3}-2a^{3}=0}$
• in the case of the angle trisection, the equation relevant here is also irreducible. ${\displaystyle x^{3}-{\frac {3}{4}}x+{\frac {a}{4}}=0}$

Both equations are of degree 3 and not of degree . ${\displaystyle 2^{k}}$

In 1845 Wantzel published a simplification of Niels Henrik Abel's proof of the impossibility of solving algebraic equations in general (and especially fifth degree) by radicals . He was referring to Abel, Joseph Liouville and Paolo Ruffini (whom, like many others, he found difficult to understand), but was above all obliged to Abel. The second part of its proof, use of the substitution theory is in the supérieure Algèbre of Serret shown

Fonts

• Research on the moyens de reconnaître si un problems de géométrie peut se résoudre avec la règle et le compas . Journal de mathématiques pures et appliquées (Liouville's Journal), Volume 2, 1837, pp. 366-372.

• Démonstration de l'impossibilité de résoudre toutes les equations algébriques avec des radicaux . Nouvelles Annales de mathématiques, Volume 4, 1845, pp. 57-65.

literature

• Adhémar Jean Claude Barré de Saint-Venant: Biography: Wantzel (PDF; 813 kB). Nouvelles Annales de mathématiques, Volume 7, 1848, pp. 321-331.
• Florian Cajori : Pierre Laurent Wantzel , Bull. Amer. Math. Soc., Vol. 24, 1918, pp. 339-347, online .

Web links

• John J. O'Connor, Edmund F. Robertson :  Pierre Wantzel. In: MacTutor History of Mathematics archive .

Individual evidence

1. ↑ For the third problem, the squaring of the circle, Ferdinand Lindemann's proof of impossibility was provided much later
2. ^ Adhémar Jean Claude Barré de Saint-Venant: Biography: Wantzel . Nouvelles Annales de mathématiques, tome 7 (1848), p. 321.
3. ↑ Bobillier is u. a. known for his work on algebraic surfaces .
4. ^ Adhémar Jean Claude Barré de Saint-Venant: Biography: Wantzel . Nouvelles Annales de mathématiques, tome 7 (1848), p. 322-324.
5. ^ Adhémar Jean Claude Barré de Saint-Venant: Biography: Wantzel . Nouvelles Annales de mathématiques, tome 7 (1848), p. 328.
6. ↑ See trigonometry formulas (section Trigonometric functions for further multiples ), ${\displaystyle \sin(3x)=3\sin x-4\sin ^{3}x\,}$
7. ↑ Liouville later (1843) published the work of Galois for the first time and recognized their importance.
8. ↑ Serret, Algèbre superieure, Volume 2, 1885, p. 512