Richard Dedekind attended school in Brunswick from the age of 7, and at this stage mathematics was not his main interest. The school was a good one and Dedekind studied 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. He entered the Collegium Carolinum at the age of 16. There he was to receive a good understanding of basic mathematics studying differential and integral calculus, analytic geometry and the foundations of analysis. He entered the University of Göttingen in 1850 with a solid grounding in mathematics.

Dedekind took a course given by Gauss on least squares. He did his doctoral work in 4 semesters under Gauss's supervision and submitted a thesis on the theory of Eulerian integrals. He received his doctorate from Göttingen in 1852, and he was to be the last pupil of Gauss. However he was not well trained in advanced mathematics and fully realized the deficiencies in his mathematical education. Dedekind therefore spent the 2 years following his doctorate learning the latest mathematical developments and working for his habilitation, which he received in 1854.

He began teaching at Göttingen giving courses on probability and geometry. He attended courses by Dirichlet on the theory of numbers, on potential theory, on definite integrals, and on partial differential equations. Dedekind and Dirichlet soon became close friends. He attended courses 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.

Dedekind started teaching at the Polytechnikum in Zürich in 1858. The Collegium Carolinum in Brunswick was upgraded to the Brunswick Polytechnikum by the 1860's, and Dedekind was appointed there in 1862. Dedekind remained there for the rest of his life, retiring in 1894. Even 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 first came to him as he was thinking about how to teach calculus. His work on mathematical induction, including the definition of finite and infinite sets, and his work in number theory, particularly in algebraic number fields, is also of major importance.

Among Dedekind's other notable contributions to mathematics were his editions of the collected works of Dirichlet, Gauss, and Riemann. His study of Dirichlet's work did, in fact, to lead to his own study of algebraic number fields, as well as to his introduction of ideals. In a joint paper with Heinrich Weber published in 1882, he applies his theory of ideals to the theory of Riemann surfaces. This gave powerful results such as a purely algebraic proof of the Riemann-Roch theorem.

Dedekind's work was quickly accepted, partly because of the clarity with which he presented his ideas. Dedekind's notion of an ideal was taken up and extended by Hilbert and then later by Emmy Noether. This led to the unique factorization of integers into powers of primes to be generalised to ideals in other rings.

In 1879 Dedekind published Über die Theorie der ganzen algebraischen Zahlen, which was again to have a large influence on the foundations of mathematics. In the book Dedekind presented a logical theory of number and of complete induction, presented his principal conception of the essence of arithmetic, and dealt with the role of the complete system of real numbers in geometry in the problem of the continuity of space.

Dedekind's legacy consisted not only of important theorems, examples, and concepts, but a whole style of mathematics that has been an inspiration to each succeeding generation. Many honours were given to Dedekind for his outstanding work, although he always remained extraordinarily modest regarding his own abilities and achievements. He was elected to the Göttingen Academy, the Berlin Academy, the Academy of Rome, the Leopoldino-Carolina Naturae Curiosorum Academia, and the Académie des Sciences in Paris. Honorary doctorates were awarded to him by the universities of Kristiania, Zürich, and Brunswick.