b. Dieuze, Moselle, France, 1822; d. Paris, 1901
Analysis (elliptic functions, differential equations);
algebra (invariant theory, quadratic forms); approximation theory
Like many who aspired to enter the École Polytechnique, Hermite undertook special preparatory classes, in his case at Lycée Henri IV and Lycée Louis-le-Grand. He began to study serious mathematics, immersing himself in the work of LAGRANGE [VI.22] and LEGENDRE [VI.24], and became interested in the solution of equations by radicals. Admitted to the École Polytechnique in 1842, by the end of that year he had completed his first significant original work. This extended results of JACOBI [VI.35] in the theory of ELLIPTIC FUNCTIONS [V.31]. He sent these to Jacobi, who responded very positively. This achievement both brought him recognition in Paris and initiated a correspondence with Jacobi on elliptic functions and number theory that launched Hermite’s career.
Hermite nonetheless struggled to find a position commensurate with his abilities, and for almost a decade survived on teaching assistant and examiner jobs around Paris. Hermite’s work turned to number theory, in particular the arithmetic of quadratic forms, where he followed GAUSS [VI.26] and Lagrange in studying when one form can be reduced to another by a linear transformation. It was in this context that the HERMITIAN MATRICES [III.50 §3] named after him arose. Hermite was interested in invariants of quadratic forms, and also applied his work to the problem of location of roots of polynomials. As a result of these efforts, in 1856 he was appointed to the Paris Academy of Sciences, with LIOUVILLE [VI.39] and CAUCHY [VI.29] supporting him. This appointment was quickly followed by Hermite’s 1858 discovery of a means to express the solutions of the general fifth-degree polynomial equation in terms of elliptic functions, which earned him widespread international recognition.
Finally obtaining a professorship at the Faculty of Science in Paris in 1869, Hermite became an influential mentor for a generation of mathematicians, his best-known protégés including J. Tannery, POINCARÉ [VI.61], E. Picard, P. Appell, and E. Goursat. Hermite’s dynastic connections are also impressive: his brother-in-law, Joseph Bertrand, was permanent secretary of the Paris Academy of Sciences, Picard was his son-in-law, Appell married Bertrand’s daughter, and their daughter married BOREL [VI.70]. His advocacy of improved international communication led to German work becoming much better known in France than it had been previously. During this period, he obtained a proof of the TRANSCENDENCE [III.41] of “e” using CONTINUEDFRACTION [III.22] methods based on earlier research in approximation theory (which had included the invention of the Hermite polynomials). His influence in the mathematical community was strong until his death.
Picard, É. 1901. L’œuvre scientifique de Charles Hermite. Annales Scientifiques de l’École Normale Supérieure (3) 18: 9-34.