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Richard Dedekind's major contribution was a redefinition of irrational numbers in terms of Dedekind cuts. He introduced the notion of an ideal which is fundamental to ring theory.Dedekind received his doctorate from Göttingen in 1852. He was the last pupil of Gauss. After taking up a chair at the University of Zürich in 1858 he returned to his home town of Brunswick in 1862 and remained there for the rest of his life.
His major contribution was a major redefinition of irrational numbers in terms of Dedekind cuts. He published this in Stetigkeit und Irrationale Zahlen in 1872.
His analysis of the nature of number and mathematical induction, including the definition of finite and infinite sets and his work in number theory, particularly in algebraic number fields, is of major importance.
In 1874 he met Cantor while on holiday in Interlaken and was sympathetic to his set theory.
Among his most 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 led to his own study of algebraic number fields, as well as his introduction of ideals.
In 1879 Dedekind published Uber die Theorie der ganzen algebraischen Zahlen 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. The general term 'ring' was introduced by Hilbert. Dedekind's notion was extended by Hilbert and Emmy Noether to allow the unique factorisation of integers into prime powers to be generalised to other rings.
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 whole new style of mathematics that been a major influence on mathematicians ever since.
References (28 books/articles)
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Tell me about Dedekind's work on set theory and his part in the history of mathematics
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