# Richard Dedekind

Julius Wilhelm Richard Dedekind ( born October 6, 1831 in Braunschweig, † February 12, 1916 ) was a German mathematician.

## Life

The son of the Brunswick lawyer and university professor Julius Dedekind visited the Martino- Katharineum Brunswick and studied mathematics from 1848 at the local Collegium Carolinum. The study, he became in 1850 continued in Göttingen, where he received his doctorate in 1852 with Carl Friedrich Gauss as his last student about the theory Eulerian integrals after only four semesters. But mathematics he heard particularly in Moritz Abraham Stern and George Ulrich at the newly established star of mathematical-physical seminar in Physics and Wilhelm Weber and Johann Benedict Listing. For Gaussian he heard in the winter semester 1850/51 on the method of least squares ( the Dedekind as one of the finest lectures kept in memory he ever heard ) and the following semester on higher geodesy. Since 1850, Dedekind was a member of the fraternity Brunsviga and held there in the summer semester of 1852, the office of Secretary and Treasurer. In 1854 he habilitated also in Göttingen, shortly after Bernhard Riemann, with whom he was friends. After the death of Gauss 1855 Peter Gustav Dirichlet was his successor and became friends with Dedekind. Dedekind in 1858 professor at the Polytechnic of Zurich, and from 1862 until his retirement in 1894, Professor of Mathematics in Braunschweig at the local Technical College. 1872 to 1875 he was its director. Although he received several calls to prestigious universities, but preferred to stay in his hometown of Braunschweig. A major reason was the close ties with his family (he had a brother and a sister, but was not married). Even after his retirement in 1894 he was still occasionally lectures. In 1859, he visited with Riemann Berlin, where he met Leopold Kronecker, Ernst Eduard Kummer and Karl Weierstrass. In 1878 he visited Paris for the Exposition.

Dedekind was a corresponding member of the Göttingen Academy of Sciences, from 1880 member of the Berlin Academy of Sciences, 1900 Corresponding Member and in 1910 foreign member of the Academie des Sciences in Paris since 1862. He was a member of the Leopoldina and the Academy in Rome. He was an honorary Doctorate in Oslo, Zurich and Brunswick.

Dedekind died on February 12, 1916, and was buried in the main cemetery Brunswick.

Dedekind played very well cello and piano and composed a chamber opera, to which his brother wrote the libretto.

## Work

Richard Dedekind was in 1888 in the Scriptures what they are and what are the numbers? the first exact implementation of the natural numbers by axioms. In his writing continuity and irrational numbers from 1872 he gave the first precise definition of the real numbers with the help of Dedekind cuts. In the Annex to number theory of his teacher Dirichlet he figured his design of the ideal theory, which was then in to the competition of Leopold Kronecker. This was the famous Supplement X in the support of Dirichlet number theory in 1871, later called Supplement XI. Named after him here are the Dedekind rings and also the Dedekind η - function in the theory of modular forms, Dedekind ζ - function of an algebraic number field, the Dedekind complementary module, Dedekind sums and the concepts of " Dedekind infinite" and " Dedekind - finite".

Dedekind played a significant role in the elaboration of abstract algebra. The algebraic concept of ring was introduced by Dedekind as well as unit and the body concept. Dedekind was also a pioneer of group theory: in his lectures 1855/56 he gave the first modern representation of the Galois theory ( which was important in addition to groups of transformations in geometry and in addition to the number theory as the third root for the formation of the group concept in the 19th century) with introduction of the abstract group concept as automorphisms of field extensions. In 1897 he led independently by George Abram Miller commutators and a Kommutatorgruppen. The concept of association also goes back to Dedekind ( 11 Supplement of Dirichlet Number Theory 1894) and Ernst Schröder end of the 19th century, but remained unnoticed at first.

He stood with Georg Cantor in the 1870s in correspondence, which is for the early history of Cantor's set theory of meaning. For example, Cantor developed as part of this exchange of letters his proof of uncountable real numbers ( letter of 7 December 1873). Both had accidentally met in 1872 in Switzerland. Their friendship ended but, after Dedekind refused to switch to Cantor at the University of Halle. Dedekind had in the 1860s counted in his algebraic work with quantities, without explicitly mentioning this and set theory used in the development of his concept of the Dedekind cut ( worked out as early as 1858 in Lectures on Analysis in Zurich ).

The stamp pictured is reminiscent of his theorem on the unique decomposability of ideals into prime ideals in the ring of integers of an algebraic number field.

He gave both the posthumous works of his teacher Dirichlet out as well as his friend Bernhard Riemann, for whose collected works he also wrote a biography. Also at the publication of the works of Carl Friedrich Gauss, he was involved.

## Quotes

In his essay What are and what are numbers? he wrote in 1888:

" What is provable, is not to be believed without proof in science. "

"The numbers are free creations of the human mind, they serve as a means to conceive the variety of things easier and sharper. Due to the purely logical structure of the science of numbers and the product obtained in their constant number imperial we are only put in a position to examine our ideas of time and space exactly by the same referring to such created in our mind Numerous. "