Lejeune Dirichlet's family came from the Belgium town of
Richelet where Dirichlet's grandfather lived. This explains the origin of his
name which comes from "Le jeune de Richelet" meaning "Young from
Richelet". Many details of the Dirichlet family are given in [6] where it
is shown that the Dirichlets came from the neighbourhood of Liège in Belgium
and not, as many had claimed, from France.
His
father was the postmaster of Düren, the town of his birth situated about
halfway between Aachen and Cologne. Even before he entered the Gymnasium in
Bonn in 1817, at the age of 12, he had developed a passion for mathematics and
spent his pocket-money on buying mathematics books. At the Gymnasium he was a
model pupil being [1]:-
... an
unusually attentive and well-behaved pupil who was particularly interested in
history as well as mathematics.
After
two years at the Gymnasium in Bonn his parents decided that they would rather
have him attend the Jesuit College in Cologne and there he had the good fortune
to be taught by Ohm. By the age of 16 Dirichlet had completed his school
qualifications and was ready to enter university. However, the standards in
German universities were not high at this time so Dirichlet decided to study in
Paris. It is interesting to note that some years later the standards in German
universities would become the best in the world and Dirichlet himself would
play a hand in the transformation.
Dirichlet
set off for France carrying with him Gauss's Disquisitiones arithmeticae
a work he treasured and kept constantly with him as others might do with the
Bible. In Paris by May 1822, Dirichlet soon contracted smallpox. It did not
keep him away from his lectures in the Collège de France and the Faculté des
Sciences for long and soon he could return to lectures. He had some of the
leading mathematicians as teachers and he was able to profit greatly from the experience
of coming in contact with Biot, Fourier, Francoeur, Hachette, Laplace, Lacroix,
Legendre, and Poisson.
From
the summer of 1823 Dirichlet was employed by General Maximilien Sébastien Foy,
living in his house in Paris. General Foy had been a major figure in the army
during the Napoleonic Wars, retiring after Napoleon's defeat at Waterloo. In
1819 he was elected to the Chamber of Deputies where he was leader of the
liberal opposition until his death. Dirichlet was very well treated by General
Foy, he was well paid yet treated like a member of the family. In return
Dirichlet taught German to General Foy's wife and children.
Dirichlet's
first paper was to bring him instant fame since it concerned the famous
Fermat's Last Theorem. The theorem claimed that for n > 2 there are
no non-zero integers x, y, z such that xn
+ yn = zn. The cases n = 3 and n
= 4 had been proved by Euler and Fermat, and Dirichlet attacked the theorem for
n = 5. Now if n = 5 then one of x, y, z is
even and one is divisible by 5. There are two cases: case 1 is when the number
divisible by 5 is even, while case 2 is when the even number and the one
divisible by 5 are distinct. Dirichlet proved case 1 and presented his paper to
the Paris Academy in July 1825. Legendre was appointed one of the referees and
he was able to prove case 2 thus completing the proof for n = 5. The
complete proof was published in September 1825. In fact Dirichlet was able to
complete his own proof of the n = 5 case with an argument for case 2
which was an extension of his own argument for case 1. It is worth noting that
Dirichlet made a later contribution proving the n = 14 case (a near miss
for the n = 7 case!).
On 28
November 1825 General Foy died and Dirichlet decided to return to Germany. He
was encouraged in this by Alexander von Humboldt who made recommendations on
his behalf. There was a problem for Dirichlet since in order to teach in a
German university he needed an habilitation. Although Dirichlet could easily
submit an habilitation thesis, this was not allowed since he did not hold a
doctorate, nor could he speak Latin, a requirement in the early nineteenth
century. The problem was nicely solved by the University of Cologne giving
Dirichlet an honorary doctorate, thus allowing him to submit his habilitation
thesis on polynomials with a special class of prime divisors to the University
of Breslau. There was, however, much controversy over Dirichlet's appointment
and the large correspondence between German professors both for and against his
appointment is considered in [15].
From
1827 Dirichlet taught at Breslau but Dirichlet encountered the same problem
which made him choose Paris for his own education, namely that the standards at
the university were low. Again with von Humboldt's help, he moved to the Berlin
in 1828 where he was appointed at the Military College. The Military College
was not the attraction, of course, rather it was that Dirichlet had an
agreement that he would be able to teach at the University of Berlin. Soon
after this he was appointed a professor at the University of Berlin where he
remained from 1828 to 1855. He retained his position in the Military College
which made his teaching and other administrative duties rather heavier than he
would have liked.
Dirichlet
was appointed to the Berlin Academy in 1831 and an improving salary from the
university put him in a position to marry, and he married Rebecca Mendelssohn,
one of the composer Felix Mendelssohn's two sisters. Dirichlet had a lifelong
friend in Jacobi, who taught at Königsberg, and the two exerted considerable
influence on each other in their researches in number theory.
In the
1843 Jacobi became unwell and diabetes was diagnosed. He was advised by his
doctor to spend time in Italy where the climate would help him recover. However,
Jacobi was not a wealthy man and Dirichlet, after visiting Jacobi and
discovering his plight, wrote to Alexander von Humboldt asking him to help
obtain some financial assistance for Jacobi from Friedrich Wilhelm IV. Dirichlet
then made a request for assistance from Friedrich Wilhelm IV, supported
strongly by Alexander von Humboldt, which was successful. Dirichlet obtained
leave of absence from Berlin for eighteen months and in the autumn of 1843 set
off for Italy with Jacobi and Borchardt. After stopping in several towns and
attending a mathematical meeting in Lucca, they arrived in Rome on 16 November
1843. Schläfli and Steiner were also with them, Schläfli's main task being to
act as their interpreter but he studied mathematics with Dirichlet as his
tutor.
Dirichlet
did not remain in Rome for the whole period, but visited Sicily and then spent
the winter of 1844/45 in Florence before returning to Berlin in the spring of
1845. Dirichlet had a high teaching load at the University of Berlin, being
also required to teach in the Military College and in 1853 he complained in a
letter to his pupil Kronecker that he had thirteen lectures a week to give in
addition to many other duties. It was therefore something of a relief when, on
Gauss's death in 1855, he was offered his chair at Göttingen.
Dirichlet
did not accept the offer from Göttingen immediately but used it to try to
obtain better conditions in Berlin. He requested of the Prussian Ministry of
Culture that he be allowed to end lecturing at the Military College. However he
received no quick reply to his modest request so he wrote to Göttingen
accepting the offer of Gauss's chair. After he had accepted the Göttingen offer
the Prussian Ministry of Culture did try to offer him improved conditions and
salary but this came too late.
The
quieter life in Göttingen seemed to suit Dirichlet. He had more time for
research and some outstanding research students. However, sadly he was not to
enjoy the new life for long. In the summer of 1858 he lectured at a conference
in Montreux but while in the Swiss town he suffered a heart attack. He returned
to Göttingen, with the greatest difficulty, and while gravely ill had the added
sadness that his wife died of a stroke.
We
should now look at Dirichlet's remarkable contributions to mathematics. We have
already commented on his contributions to Fermat's Last Theorem made in 1825. Around
this time he also published a paper inspired by Gauss's work on the law of
biquadratic reciprocity. Details are given in [13] where Rowe discusses the
importance of the intellectual and personal relationship between Gauss and
Dirichlet.
He
proved in 1837 that in any arithmetic progression with first term coprime to
the difference there are infinitely many primes. This had been conjectured by
Gauss. Davenport wrote in 1980 (see [16]):-
Analytic
number theory may be said to begin with the work of Dirichlet, and in
particular with Dirichlet's memoir of 1837 on the
existence of primes in a given arithmetic progression.
Shortly
after publishing this paper Dirichlet published two further papers on analytic
number theory, one in 1838 with the next in the following year. These papers
introduce Dirichlet series and determine, among other things, the formula for
the class number for quadratic forms.
His
work on units in algebraic number theory Vorlesungen über Zahlentheorie
(published 1863) contains important work on ideals. He also proposed in 1837
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
mechanics he investigated the equilibrium of systems and potential theory. These
investigations began in 1839 with papers which gave methods to evaluate
multiple integrals and he applied this to the problem of the gravitational
attraction of an ellipsoid on points both inside and outside. He turned to
Laplace's problem of proving the stability of the solar system and produced an
analysis which avoided the problem of using series expansion with quadratic and
higher terms disregarded. This work led him to the Dirichlet problem concerning
harmonic functions with given boundary conditions. Some work on mechanics later
in his career is of quite outstanding importance. In 1852 he studied the
problem of a sphere placed in an incompressible fluid, in the course of this
investigation becoming the first person to integrate the hydrodynamic equations
exactly.
Dirichlet
is also well known for his papers on conditions for the convergence of
trigonometric series and the use of the series to represent arbitrary
functions. These series had been used previously by Fourier in solving
differential equations. Dirichlet's work is published in Crelle's Journal
in 1828. Earlier work by Poisson on the convergence of Fourier series was shown
to be non-rigorous by Cauchy. Cauchy's work itself was shown to be in error by
Dirichlet who wrote of Cauchy's paper:-
The
author of this work himself admits that his proof is defective for certain
functions for which the convergence is, however, incontestable.
Because
of this work Dirichlet is considered the founder of the theory of Fourier
series. Riemann, who was a student of Dirichlet, wrote in the introduction to
his habilitation thesis on Fourier series that it was Dirichlet [11]:-
...
who wrote the first profound paper about the subject.
In [1]
Dirichlet's character and teaching qualities are summed up as follows:-
He was
an excellent teacher, always expressing himself with great clarity. His manner
was modest; in his later years he was shy and at times reserved. He seldom
spoke at meetings and was reluctant to make public appearances.
At age
45 Dirichlet was described by Thomas Hirst as follows:-
He is
a rather tall, lanky-looking man, with moustache and beard about to turn grey
with a somewhat harsh voice and rather deaf. He was unwashed, with his cup of
coffee and cigar. One of his failings is forgetting time, he pulls his watch
out, finds it past three, and runs out without even finishing the sentence.
Koch,
in [11], sums up Dirichlet's contribution writing that:-
...
important parts of mathematics were influenced by Dirichlet. His proofs
characteristically started with surprisingly simple observations, followed by
extremely sharp analysis of the remaining problem. With Dirichlet began the
golden age of mathematics in Berlin.
Article
by: J J O'Connor and E F Robertson
May 2000
MacTutor History of Mathematics
[http://www-history.mcs.st-andrews.ac.uk/Mathematicians/Dirichlet.html]