Johann Peter Gustav Lejeune Dirichlet Biography

Johann Peter Gustav Lejeune Dirichlet was born in 1805, the son of the town postmaster of Düren (then part of the French empire). He was initially educated at public schools, then at a private school which stressed Latin. Interest in mathematics surfaced early for Dirichlet and by age 12 he was purchasing mathematics textbooks. Dirichlet enrolled at Bonn's Gymnasium in 1817, where he showed great interest in mathematics and history.

Two years later, Dirichlet's parents sent him to a Jesuit college in Cologne. He was a student of physicist George Simon Ohm, under whom he received thorough training in theoretical physics. At the young age of 16, Dirichlet completed his Abitur examination. His parents wanted him to study law, but he was already well on his way in the field of mathematics.

Other than Karl Gauss, Germany, at that time, had no notable mathematicians. Paris, on the other hand, boasted such luminaries as Pierre Simon Laplace, Adrien-Marie Legendre, and Jospeh Fourier. In 1822, Dirichlet visited Paris. He was not there long before he caught a mild case of smallpox, but it was not severe enough to keep him from continuing classes at the College de France, and the Faculte des Sciences. In 1823, he was appointed to a well-paid position as a tutor to General Maximilian Fay's children. Fay was a national hero of the Napoleonic wars and a liberal opposition leader in the Chamber of Deputies. Dirichlet was treated as a member of the family, thus meeting prominent French intellectuals, including mathematician Joseph Fourier. Fourier's ideas influenced Dirichlet's later works on trigonometric seriesand mathematical physics.

Dirichlet's main interest was number theory, which was first ignited through an early study of Gauss' Disquistiones arthmeticae (1801). In June 1825, Dirichlet presented his first paper on mathematics, "Memoire sur l'impossibilite de quelques equations indeterminees du cinquieme degre," to the French Académie Royale des Sciences. The paper explored Diophantine equationsof the form x⁵ + y⁵ = Az⁵ using algebraic number theory, Dirichlet's favorite area of study. Legendre extended these results to give a proof of Fermat's last theorem for n=5.

With General Fay's death in 1825, Dirichlet returned to Germany. Fellow German scientist Alexander von Humboldt strongly supported Dirichlet's return, as Germany needed strengthening in the natural sciences. Although Dirichlet did not have the required doctorate, he was permitted to qualify at the University of Breslau for the habilitation required to teach at a German university.

Breslau was not an inspiring environment for scientific work, so in 1828, again with Humbolt's assistance, Dirichlet moved to Berlin and began teaching mathematics at the military academy. At age 23, he was first appointed to a temporary position at the University of Berlin and in 1831 became a member of the Berlin Academy of Sciences. That same year Dirichlet married Rebecca Mendelssohn-Bartholdy, granddaughter of philosopher Moses Mendelssohn, and sister to composer Felix Mendelssohn. In 1832, he published a proof of Pierre de Fermat's Last Theorem for n=14.

During his 27 years as professor in Berlin, Dirichlet influenced the development of German mathematics through his lectures, pupils, and scientific papers. He taught with great clarity and his published scientific papers were of the highest quality.

Dirichlet was a shy, modest man, who rarely made public appearances, or spoke at meetings. His lifelong friend was mathematician Karl Jacobi. Both mathematicians influenced each other's work, particularly in number theory. In 1843, Jacobi moved to Rome for health reasons. This prompted Dirichlet to request a leave of absence to move his family to Rome with Jacobi. Dirichlet remained in Italy for a year and a half, visited Sicily, and spent the winter in Florence.

At a meeting of the Academy of Science, held on July 27, 1837, Dirichlet presented a paper on analytic number theory. The paper offers proof of the fundamental theorem that bears his name: any arithmetic sequence of integers: an + b, n = 0, 1, 2 . . ., where a and b are relatively prime, must include an infinite number of primes. This paper was followed in 1838 and 1839 by a two-part paper on analytic number theory, "Recherches sur diverses applications de l'analyse infinitesimale a la theorie des nombres." After publication of his fundamental papers, the importance of his number theory work declined, but Dirichlet continued to publish papers in other areas.

Dirichlet is best known for his work on trigonometric series and mathematical physics. In an 1828 paper in Crelle's Journal, he gave the first rigorous proof of sufficient conditions for the convergence of the Fourier seriesfor a function. His investigations of equilibrium of systems and potential theory gave rise to what is now called the Dirichlet problemabout formulating and solving a class of partial differential equations that arise from the flow of heat, electricity, and fluids subject to given boundary conditions.

In 1837 Dirichlet proposed the modern definition of a function: if a variable y is so related to a variable x that whenever a numerical value is assigned to x, there is a rule according to which a unique value of y is determined, then y is said to be a function of the independent variable x. In another 1837 paper, Dirichlet proved that in an absolutely convergent series, one may rearrange the order in which terms are added in whatever way one wishes and not change the sum of the series. While this is immediate for a finite sum, the fact that an infinite sum of numbers could always be rearranged without changing the value was surprising. He also gave examples of conditionally convergent series in which the sum wasaltered by rearrangement of the terms. Almost 20 years later, Georg Riemann proved that the terms of a conditionally convergent series could be rearranged to yield a sum of any desired value.

At the golden jubilee celebration of his doctorate, Dirichlet's teacher Karl Gauss tried to light his pipe with a piece of his original manuscript Disquisitiones arithmeticae. Dirichlet was overcome by the sacrilege of such an action. He rescued the piece of paper from Gauss' fire. Dirichlet treasured the paper for his remaining years, and his editors found it among his papers after his death.

With Gauss's death in 1855, the University of Göttingen sought a successor of great distinction and chose Dirichlet. His current position at the military academy was unappealing and lacked scientific stimulation and he was required to lecture 13 times a week. Dirichlet accepted the university's offer.

Dirichlet moved to Göttingen in 1855, where he purchased a house with a garden. He enjoyed a quiet life there with excellent students and the time available for research. But his new contentment did not last long. During a speech in Montreaux, Switzerland, he suffered a heart attack and barely made it home. During his illness, his wife died of a stroke, and Dirichlet subsequently died the following spring in 1863.

After his death, Dirichlet's pupil and friend, Julius Dedekind, published Dirichlet's Vorlesungen über Zahlentheorie, adding several supplements of his own investigations on algebraic number theory. The addenda are regarded as one of the most important sources for the creation of the theory of ideals, and are the core of algebraic number theory.