Peter Gustav Lejeune Dirichlet facts for kids

Quick facts for kids Peter Gustav Lejeune Dirichlet
Born	Johann Peter Gustav Lejeune Dirichlet (1805-02-13)13 February 1805 Düren, French Empire
Died	5 May 1859(1859-05-05) (aged 54) Göttingen, Kingdom of Hanover
Nationality	German
Known for	See full list
Awards	PhD (Hon): University of Bonn (1827) Pour le Mérite (1855)
Scientific career
Fields	Mathematician
Institutions	University of Breslau University of Berlin University of Göttingen
Thesis	Partial Results on Fermat's Last Theorem, Exponent 5 (1827)
Academic advisors	Siméon Poisson Joseph Fourier Carl Gauss
Doctoral students	Gotthold Eisenstein Leopold Kronecker Rudolf Lipschitz Carl Wilhelm Borchardt
Other notable students	Moritz Cantor Elwin Bruno Christoffel Richard Dedekind Alfred Enneper Eduard Heine Bernhard Riemann Ludwig Schläfli Ludwig von Seidel Wilhelm Weber Julius Weingarten

Johann Peter Gustav Lejeune Dirichlet (born February 13, 1805 – died May 5, 1859) was an important German mathematician. He made big contributions to number theory, which is a branch of mathematics that studies whole numbers and their properties. He also helped create a new field called analytic number theory. Dirichlet also worked on Fourier series and other areas of mathematical analysis. He is known for being one of the first to clearly define what a function is in modern mathematics.

Even though his full last name was Lejeune Dirichlet, people usually called him Dirichlet. Many mathematical ideas and results are named after him.

Discovering Mathematics: Dirichlet's Early Life

Growing Up: Dirichlet's Childhood and Education

Gustav Lejeune Dirichlet was born on February 13, 1805, in Düren. This town was then part of the First French Empire. Later, in 1815, it became part of Prussia. His father, Johann Arnold Lejeune Dirichlet, worked as a postmaster and was also a city council member. The family name "Lejeune Dirichlet" came from his grandfather, who moved to Düren from a small place called Richelette in Belgium. The name means "the youth from Richelette" in French.

Dirichlet was the youngest of seven children, and his family was not rich. However, his parents supported his education. They first sent him to elementary school and then to a private school. They hoped he would become a merchant. But young Dirichlet loved mathematics. By age 12, he convinced his parents to let him continue studying math.

In 1817, he went to the Beethoven-Gymnasium in Bonn. In 1820, Dirichlet moved to the Jesuit Gymnasium in Cologne. There, his lessons with Georg Ohm helped him learn even more about mathematics. He left the school a year later with only a certificate. He couldn't speak Latin well enough to earn a full diploma.

Studying in Paris: A Young Mathematician's Journey

Dirichlet again persuaded his parents to support his math studies. They wanted him to study law, but he was set on mathematics. At that time, there were not many chances to study advanced mathematics in Germany. So, Dirichlet decided to go to Paris in May 1822.

In Paris, he took classes at the Collège de France and the University of Paris. He learned from mathematicians like Hachette. He also studied Gauss's book, Disquisitiones Arithmeticae, on his own. He kept this book with him his whole life. In 1823, he found a job teaching German to the children of General Maximilien Foy. This job finally allowed him to support himself.

Dirichlet's first important research made him famous right away. He worked on Fermat's Last Theorem for the case where n equals 5. This was the first progress on the theorem since Fermat himself proved the case for n equals 4. Another mathematician, Adrien-Marie Legendre, soon finished the proof for this case. Dirichlet completed his own proof shortly after. A few years later, he fully proved the case for n equals 14. In June 1825, when he was only 20 and had no degree, he was allowed to present his work on the n equals 5 case at the French Academy of Sciences. This was a huge achievement! His presentation also connected him with Fourier and Poisson. They sparked his interest in theoretical physics, especially Fourier's ideas about heat.

Dirichlet's Career and Family Life

Returning to Prussia: Teaching in Breslau

General Foy died in November 1825, and Dirichlet could not find another job in France. So, he had to go back to Prussia. Fourier and Poisson introduced him to Alexander von Humboldt, a famous scientist. Humboldt helped Dirichlet by writing letters to the Prussian government and the Prussian Academy of Sciences. He also got a strong recommendation from Gauss, who praised Dirichlet's "excellent talent."

With this support, Dirichlet was offered a teaching job at the University of Breslau. He hadn't finished his doctoral degree, so he used his work on Fermat's theorem as his thesis for the University of Bonn. Again, his poor Latin skills made it hard for him to defend his thesis publicly. After much discussion, the university gave him an honorary doctorate in February 1827. He also got special permission to skip the Latin exam needed for teaching. Dirichlet became a lecturer at Breslau for the 1827–28 school year.

While in Breslau, Dirichlet continued his research in number theory. He published important work on the biquadratic reciprocity law, which Gauss was also studying. Alexander von Humboldt used these new results to help Dirichlet move to Berlin. Because Dirichlet was only 23, Humboldt could only get him a trial position at the Prussian Military Academy in Berlin. He was still officially employed by the University of Breslau. This trial period lasted three years, and his position became permanent in 1831.

Marriage and Family: Rebecka Mendelssohn

Dirichlet married Rebecka Mendelssohn in 1832. They had two children, Walter (born 1833) and Flora (born 1845). This drawing of Rebecka was made by Wilhelm Hensel in 1823.

After moving to Berlin, Dirichlet met the famous Mendelssohn family through Alexander von Humboldt. Their home was a popular meeting place for artists and scientists in Berlin. Among them were Abraham Mendelssohn Bartholdy's children, Felix and Fanny Mendelssohn, who were both amazing musicians. The painter Wilhelm Hensel (Fanny's husband) was also often there. Dirichlet became very interested in Abraham's daughter, Rebecka, and they married in 1832.

Rebecka Henriette Lejeune Dirichlet (born Rebecka Mendelssohn) was born on April 11, 1811, in Hamburg. She was the youngest sister of Felix and Fanny Mendelssohn and the granddaughter of Moses Mendelssohn, a famous philosopher. Rebecka was part of her parents' well-known social gatherings, where important musicians, artists, and scientists met. In 1833, their first son, Walter, was born. Rebecka died in Göttingen in 1858.

Life in Berlin: Teaching and Research

As soon as he arrived in Berlin, Dirichlet wanted to teach at the University of Berlin. The Education Minister approved his transfer in 1831. However, he still needed to complete a formal qualification. Even though he wrote a paper for it, he put off giving the required Latin lecture for 20 more years, until 1851. Because of this, he didn't have full rights as a professor and earned less money. This meant he had to keep his teaching job at the Military School. In 1832, Dirichlet became the youngest member of the Prussian Academy of Sciences at just 27 years old.

Dirichlet was known for explaining things clearly, and students liked his teaching. He especially enjoyed lecturing on advanced topics he was researching, like number theory, analysis, and mathematical physics. He was the first German professor to teach number theory. He guided the doctoral studies of several important German mathematicians, including Gotthold Eisenstein, Leopold Kronecker, and Rudolf Lipschitz. He also influenced many other scientists. At the Military Academy, Dirichlet introduced differential and integral calculus into the lessons, which improved the science education there. However, he started to feel that teaching at both the Military Academy and the university took too much time away from his own research.

While in Berlin, Dirichlet stayed in touch with other mathematicians. In 1829, he met Carl Jacobi, a math professor. They became close friends and often discussed their research. In 1839, Dirichlet met Joseph Liouville in Paris, and they also became friends. In 1839, Jacobi sent Dirichlet a paper by Ernst Kummer, who was a schoolteacher at the time. Dirichlet and Jacobi saw Kummer's talent and helped him get elected to the Berlin Academy. In 1842, they helped him get a full professor position at the University of Breslau. Kummer later married Ottilie Mendelssohn, who was Rebecka's cousin.

In 1843, when Jacobi became ill, Dirichlet went to help him. He even arranged for the King's personal doctor to see Jacobi. When the doctor suggested Jacobi go to Italy, Dirichlet and his family joined him on the trip. They were joined by Ludwig Schläfli, who helped as a translator. Schläfli was very interested in math, and Dirichlet and Jacobi taught him during the trip. He later became an important mathematician himself. The Dirichlet family stayed in Italy until 1845, and their daughter Flora was born there. In 1844, Jacobi moved to Berlin, and their friendship grew even stronger. In 1846, when another university tried to hire Dirichlet, Jacobi helped Humboldt get Dirichlet's pay doubled to keep him in Berlin. Still, he wasn't paid as much as a full professor and couldn't leave the Military Academy.

Dirichlet and his family held liberal views and supported the 1848 revolution. He even guarded the Prince of Prussia's palace with a rifle. After the revolution failed, the Military Academy closed temporarily, which meant he lost a lot of income. When it reopened, the atmosphere became difficult for him because the officers he taught were expected to be loyal to the government. Some newspapers that didn't support the revolution criticized him and other liberal professors.

In 1849, Dirichlet and his friend Jacobi celebrated Gauss's doctorate anniversary.

Moving to Göttingen: Final Years

Even though Dirichlet was very skilled and received many honors, and even though he finally met all the requirements for a full professor in 1851, his pay at the university remained low. He still couldn't leave the Military Academy. In 1855, after Gauss died, the University of Göttingen invited Dirichlet to take Gauss's place. Because of the problems he faced in Berlin, he decided to accept the offer and moved to Göttingen with his family. Ernst Kummer took over his position as a math professor in Berlin.

Dirichlet enjoyed his time in Göttingen. He had less teaching to do, which gave him more time for research. He also worked closely with a new generation of researchers, especially Richard Dedekind and Bernhard Riemann. After moving to Göttingen, he helped Riemann get a small yearly payment to stay on the teaching staff there. Dedekind, Riemann, Moritz Cantor, and Alfred Enneper all had their PhDs already, but they still attended Dirichlet's classes to learn from him. Dedekind felt that studying with Dirichlet made him "a new human being." He later edited and published Dirichlet's lectures and other work on number theory in a book called Vorlesungen über Zahlentheorie (Lectures on Number Theory).

In the summer of 1858, while on a trip, Dirichlet had a heart attack. He died in Göttingen on May 5, 1859, a few months after his wife Rebecka passed away. Dirichlet's brain is kept at the University of Göttingen, along with Gauss's brain. The Academy in Berlin honored him with a special speech by Kummer in 1860. Later, they arranged for his collected works to be published.

Dirichlet's Mathematical Discoveries

Contributions to Number Theory

Number theory was Dirichlet's main area of interest. He made several important discoveries and created new tools to prove them. Many of these tools are now named after him. In 1837, he published Dirichlet's theorem on arithmetic progressions. This theorem used ideas from mathematical analysis to solve a problem in algebra. This led to the creation of a new field called analytic number theory. To prove this theorem, he introduced Dirichlet characters and L-functions. He also pointed out the difference between absolute and conditional convergence of series.

In 1838 and 1839, he proved the first class number formula for quadratic forms. This formula was later improved by his student Kronecker. This discovery opened the door for similar results in other areas of number theory. Based on his study of how units (special numbers) are structured in quadratic fields, he proved the Dirichlet unit theorem. This is a very important result in algebraic number theory.

He was also the first to use the pigeonhole principle, a simple counting method, to prove a theorem in diophantine approximation. This theorem is now called Dirichlet's approximation theorem. He made important contributions to Fermat's Last Theorem, proving the cases for n equals 5 and n equals 14. He also worked on the biquadratic reciprocity law. The Dirichlet divisor problem, which he first studied, is still an unsolved problem in number theory today.

Defining the Modern Concept of a Function

Dirichlet helped define what a function is in modern mathematics. He said that for any x, there is only one y that corresponds to it. This was different from older, less clear ideas of what a function was. He focused on functions that are continuous in pieces. Some historians say he introduced the modern idea of a function. However, others argue that he still thought of functions as having two values at points where they were not continuous.

Other Areas of Math and Physics

Dirichlet also worked in mathematical physics. He gave lectures and published research on potential theory, which includes the Dirichlet problem and Dirichlet principle. He also studied the theory of heat and hydrodynamics. He improved on Lagrange's work on conservative systems. He showed that for a system to be in balance, its potential energy must be as low as possible.

Dirichlet also taught about probability theory and least squares. He developed new methods and results, especially for limit theorems. He also improved Laplace's method for approximations related to the central limit theorem. The Dirichlet distribution and the Dirichlet process, which are based on the Dirichlet integral, are named after him.

Honors and Recognition

Dirichlet was chosen as a member of several important academies:

• Prussian Academy of Sciences (1832)
• Saint Petersburg Academy of Sciences (1833) – corresponding member
• Göttingen Academy of Sciences (1846)
• French Academy of Sciences (1854) – foreign member
• Royal Swedish Academy of Sciences (1854)
• Royal Belgian Academy of Sciences (1855)
• Royal Society (1855) – foreign member

In 1855, Dirichlet received the Pour le Mérite medal, a high honor, thanks to Alexander von Humboldt's recommendation. There is a crater on the Moon and an asteroid, 11665 Dirichlet, named after him.

Selected Publications

• Lejeune Dirichlet, J.P.G. (1889). L. Kronecker. ed. Werke. 1. Berlin: Reimer.
• Lejeune Dirichlet, J.P.G. (1897). L. Kronecker, L. Fuchs. ed. Werke. 2. Berlin: Reimer.
• Lejeune Dirichlet, J.P.G.; Richard Dedekind (1863). Vorlesungen über Zahlentheorie. F. Vieweg und sohn.

Images for kids

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  Drawing of Rebecka Mendelssohn by Wilhelm Hensel, 1823.

See also

In Spanish: Peter Gustav Lejeune Dirichlet para niños