Peter Gustav Lejeune Dirichlet

Lore & Background

Dirichlet was born on 13 February 1805 in Düren, a town on the left bank of the Rhine then part of the First French Empire. His father, Johann Arnold Lejeune Dirichlet, was the postmaster, merchant, and city councilor. The family surname derived from Richelette (or Richelle), a small community 5 km northeast of Liège in Belgium. Despite being the youngest of seven children and not wealthy, his parents supported his education after he showed a strong interest in mathematics before age 12. He studied at Gymnasium Bonn and later the Jesuit Gymnasium in Cologne, where lessons with Georg Ohm widened his mathematical knowledge. He left gymnasium without the Abitur due to his inability to speak fluent Latin. His first original research, a partial proof of Fermat's Last Theorem for n=5, brought him immediate fame. Adrien-Marie Legendre soon completed the proof for this case; Dirichlet completed his own proof a short time after Legendre, and a few years later produced a full proof for n=14. In June 1825 he lectured on his partial proof at the French Academy of Sciences, an exceptional feat for a 20-year-old student with no degree. Through Fourier and Poisson, he became interested in theoretical physics. After General Foy died in November 1825, Dirichlet returned to Prussia. With support from Alexander von Humboldt and Gauss, he was offered a teaching position at the University of Breslau. He submitted his Fermat memoir as a thesis to the University of Bonn, which awarded him an honorary doctorate in February 1827. He earned the Habilitation and lectured as a Privatdozent at Breslau in the 1827–28 year. While there, he published important contributions to the biquadratic reciprocity law. Humboldt used these results, praised by Friedrich Bessel, to arrange his transfer to Berlin. At age 23, he got a trial position at the Prussian Military Academy, which became permanent in 1831. In Berlin, Humboldt introduced Dirichlet to the salons of banker Abraham Mendelssohn Bartholdy. Dirichlet married Abraham's daughter Rebecka in 1832. Rebecka was a granddaughter of Moses Mendelssohn and the youngest sister of Felix Mendelssohn and Fanny Mendelssohn. She was born in Hamburg on 11 April 1811 and died in Göttingen on 1 December 1858. Their first son, Walter, was born in 1833. Dirichlet applied to lecture at the University of Berlin, and the Education Minister approved the transfer in 1831. The faculty required a renewed habilitation; Dirichlet wrote a Habilitationsschrift but postponed the mandatory Latin lecture until 1851. He remained attached to the faculty with less than full rights, forcing him to keep his teaching position at the Military School. In 1832 he became a member of the Prussian Academy of Sciences, the youngest member at 27. He advised doctoral theses of Gotthold Eisenstein, Leopold Kronecker, Rudolf Lipschitz, and Carl Wilhelm Borchardt, and influenced many others including Elwin Bruno Christoffel, Wilhelm Weber, Eduard Heine, Ludwig von Seidel, and Julius Weingarten. At the Military Academy, he introduced differential and integral calculus into the curriculum. While in Berlin, Dirichlet kept in contact with other mathematicians. In 1829 he met Carl Jacobi; they became close friends. In 1839, during a visit to Paris, he met Joseph Liouville; they became friends and visited each other with their families. In 1839, Jacobi sent Dirichlet a paper by Ernst Kummer, then a schoolteacher. Realizing Kummer's potential, they helped him get elected to the Berlin Academy and, in 1842, obtained a full professor position for him at the University of Breslau. In 1840 Kummer married Ottilie Mendelssohn, a cousin of Rebecka.

Reader's Guide

Dirichlet's significance lies in his creation of analytic number theory, which uses analysis to study number-theoretic problems. His work on Fermat's Last Theorem for n=5 and n=14 marked the first advances since Fermat's proof for n=4 and Euler's proof for n=3. In analysis, he advanced the theory of Fourier series and was one of the first to give the modern formal definition of a function. In mathematical physics, he studied potential theory, boundary-value problems, heat diffusion, and hydrodynamics. His teaching at Berlin influenced a generation of German mathematicians. He was the first German professor to give lectures on number theory. His clarity in lectures made him highly regarded by students. He advised the doctoral theses of Gotthold Eisenstein, Leopold Kronecker, Rudolf Lipschitz, and Carl Wilhelm Borchardt, and influenced many others including Elwin Bruno Christoffel, Wilhelm Weber, Eduard Heine, Ludwig von Seidel, and Julius Weingarten. At the Military Academy, he introduced differential and integral calculus into the curriculum, raising the level of scientific education there. Dirichlet's legacy endures in both pure and applied mathematics. The Dirichlet boundary condition and Dirichlet function are among the many concepts named after him. His work on the biquadratic reciprocity law contributed to a focal point of Gauss's research. His personal life was intertwined with the Mendelssohn family; his wife Rebecka was a granddaughter of Moses Mendelssohn and sister of Felix and Fanny Mendelssohn. She sang a small role in the 1829 premiere of Felix's Singspiel *Die Heimkehr aus der Fremde* at the Mendelssohn house.

Did You Know?

• Dirichlet was born on 13 February 1805 in Düren, a town on the left bank of the Rhine that was then part of the First French Empire.
• His surname 'Lejeune Dirichlet' comes from 'le jeune de Richelette' (French for 'the youth from Richelette'), referring to his paternal grandfather's origin in Richelette (or Richelle), a small community 5 km northeast of Liège in Belgium.
• At age 20, Dirichlet lectured on his partial proof of Fermat's Last Theorem for n=5 at the French Academy of Sciences, an exceptional feat for a student with no degree.
• Dirichlet married Rebecka Mendelssohn in 1832; she was the youngest sister of composer Felix Mendelssohn and a granddaughter of philosopher Moses Mendelssohn.
• Dirichlet was the first German professor to give lectures on number theory at a university.