**Richard Dedekind**'s major contribution was a redefinition of irrational numbers in terms of Dedekind cuts. He introduced the notion of an ideal in Ring Theory.

- The school, Martino-Catharineum, was a good one and Dedekind studied science, in particular physics and chemistry.
- However, physics became less than satisfactory to Dedekind with what he considered an imprecise logical structure and his attention turned towards mathematics.
- The two departments combined to initiate a seminar which Dedekind joined from its beginning.
- The first course to really make Dedekind enthusiastic was, rather surprisingly, a course on experimental physics taught by Weber.
- More likely it was Weber who inspired Dedekind rather than the topic of the course.
- In the autumn term of 1850, Dedekind attended his first course given by Gauss.
- fifty years later Dedekind remembered the lectures as the most beautiful he had ever heard, writing that he had followed Gauss with constantly increasing interest and that he could not forget the experience.
- Dedekind did his doctoral work in four semesters under Gauss's supervision and submitted a thesis on the theory of Eulerian integrals.
- At this time Berlin was the place where courses were given on the latest mathematical developments but Dedekind had not been able to learn such material at Göttingen.
- Dedekind therefore spent the two years following the award of his doctorate learning the latest mathematical developments and working for his habilitation.
- In 1854 both Riemann and Dedekind were awarded their habilitation degrees within a few weeks of each other.
- Dedekind was then qualified as a university teacher and he began teaching at Göttingen giving courses on probability and geometry.
- This was an extremely important event for Dedekind who found working with Dirichlet extremely profitable.
- Dedekind and Dirichlet soon became close friends and the relationship was in many ways the making of Dedekind, whose mathematical interests took a new lease of life with the discussions between the two.
- recalled in later years that he only knew Dedekind by sight because Dedekind always arrived and left with Dirichlet and was completely eclipsed by him.
- Dedekind certainly still continued to learn mathematics at this time as a student would by attending courses, such as those by Riemann on abelian functions and elliptic functions.
- Around this time Dedekind studied the work of Galois and he was the first to lecture on Galois theory when he taught a course on the topic at Göttingen during this period.
- While at Göttingen, Dedekind applied for J L Raabe's chair at the Polytechnikum in Zürich.
- Dirichlet supported his application writing that Dedekind was 'an exceptional pedagogue'.
- In the spring of 1858 the Swiss councillor who made appointments came to Göttingen and Dedekind was quickly chosen for the post.
- Dedekind was appointed to the Polytechnikum in Zürich and began teaching there in the autumn of 1858.
- It was while he was thinking how to teach differential and integral calculus, the first time that he had taught the topic, that the idea of a Dedekind cut came to him.
- Dedekind's brilliant idea was to represent the real numbers by such divisions of the rationals.
- Dedekind and Riemann travelled together to Berlin in September 1859 on the occasion of Riemann's election to the Berlin Academy of Sciences.
- In Berlin, Dedekind met Weierstrass, Kummer, Borchardt and Kronecker.
- The Collegium Carolinum in Brunswick had been upgraded to the Brunswick Polytechnikum by the 1860s, and Dedekind was appointed to the Polytechnikum in 1862.
- Dedekind remained there for the rest of his life, retiring on 1 April 1894.
- After he retired, Dedekind continued to teach the occasional course and remained in good health in his long retirement.
- Dedekind made a number of highly significant contributions to mathematics and his work would change the style of mathematics into what is familiar to us today.
- One remarkable piece of work was his redefinition of irrational numbers in terms of Dedekind cuts which, as we mentioned above, first came to him as early as 1858.
- Dedekind loved to take his holidays in Switzerland, the Austrian Tyrol or the Black Forest in southern Germany.
- In this quote Dedekind is arguing against Kronecker's objections to the infinite and, therefore, is agreeing with Cantor's views.
- Among Dedekind's other notable contributions to mathematics were his editions of the collected works of Peter Dirichlet, Carl Gauss, and Georg Riemann.
- Dedekind's study of Dirichlet's work did, in fact, lead to his own study of algebraic number fields, as well as to his introduction of ideals.
- Dedekind edited Dirichlet's lectures on number theory and published these as Vorlesungen über Zahlentheorie Ⓣ(Lectures on Number Theory) in 1863.
- It was in the third and fourth editions of Vorlesungen über Zahlentheorie Ⓣ(Lectures on Number Theory), published in 1879 and 1894, that Dedekind wrote supplements in which he introduced the notion of an ideal which is fundamental to ring theory.
- Dedekind formulated his theory in the ring of integers of an algebraic number field.
- Dedekind, in a joint paper with Heinrich Weber published in 1882, applies his theory of ideals to the theory of Riemann surfaces.
- Dedekind's work was quickly accepted, partly because of the clarity with which he presented his ideas and partly since Heinrich Weber lectured to Hilbert on these topics at the University of Königsberg.
- Dedekind's notion of ideal was taken up and extended by Hilbert and then later by Emmy Noether.
- In 1879 Dedekind published Über die Theorie der ganzen algebraischen Zahlen Ⓣ(On the theory of algebraic integers) which was again to have a large influence on the foundations of mathematics.
- Dedekind's brilliance consisted not only of the theorems and concepts that he studied but, because of his ability to formulate and express his ideas so clearly, he introduced a new style of mathematics that been a major influence on mathematicians ever since.
- Many honours were given to Dedekind for his outstanding work, although he always remained extraordinarily modest regarding his own abilities and achievements.

Born 6 October 1831, Braunschweig, duchy of Braunschweig (now Germany). Died 12 February 1916, Braunschweig, duchy of Braunschweig (now Germany).

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Algebra, Analysis, Group Theory, Origin Germany, Set Theory

**O’Connor, John J; Robertson, Edmund F**: MacTutor History of Mathematics Archive