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Encyclopedia > Ernst Kummer

## Contributions to mathematics

Kummer made several contributions to mathematics in different areas; he codified some of the relations between different hypergeometric series (contiguity relations). The Kummer surface results from taking the quotient of a two-dimensional abelian variety by the cyclic group {1, −1} (an early orbifold: it has 16 singular points, and its geometry was intensively studied in the nineteenth century). See also Kummer's function. In mathematics, a hypergeometric series is a power series in which the ratios of successive coefficients k is a rational function of k. ... A K3 manifold is a hyperkähler manifold of real dimension 4, i. ... In mathematics, particularly in algebraic geometry, complex analysis and number theory, abelian variety is a term used to denote a complex torus that can be embedded into projective space as a projective variety. ... In topology, an orbifold is a generalization of manifold. ... Alternative meaning: Nineteenth Century (periodical) (18th century — 19th century — 20th century — more centuries) As a means of recording the passage of time, the 19th century was that century which lasted from 1801-1900 in the sense of the Gregorian calendar. ... In mathematics, there are several functions known as Kummers function. ...

## Kummer and Fermat's last theorem

Kummer also proved Fermat's last theorem for a considerable class of prime exponents (see regular prime, ideal class group). His methods were closer, perhaps, to p-adic ones than to ideal theory as understood later, though the term 'ideal' arose here. He studied what were later called Kummer extensions of fields: that is, extensions generated by adjoining an nth root to a field already containing a primitive nth root of unity. This is a significant extension of the theory of quadratic extensions, and the genus theory of quadratic forms (linked to the 2-torsion of the class group). As such, it is still foundational for class field theory. Pierre de Fermat Fermats last theorem (sometimes abbreviated as FLT and also called Fermats great theorem) is one of the most famous theorems in the history of mathematics. ... In mathematics, regular primes are a certain kind of prime numbers. ... In mathematics the theory of algebraic number fields gives rise to a finite abelian group constructed from each such field, its ideal class group. ... The p-adic number systems were first described by Kurt Hensel in 1897. ... In ring theory, a branch of abstract algebra, an ideal is a special subset of a ring which generalizes important properties of integers. ... In mathematics, a Kummer extension of fields is a field extension L/K where for some given integer n > 1 we have [L:K] = n and L is generated over K by a root of a polynomial Xn − a with a in K, and K contains n distinct roots of... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ... In mathematics, the nth roots of unity or de Moivre numbers are all the complex numbers which yield 1 when raised to a given power n. ... In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables. ... In mathematics, class field theory is a major branch of algebraic number theory. ...

## Proof of an infinite number of primes

In 1878 Kummer gave this proof that there are an infinite number of primes.

Assume the number of primes is finite: p1, p2, ..., pk. Let n be the product of all of these primes. Now n–1 is not prime because it is greater than pk. If n–1 is not prime then there is some i such that pi divides n–1. But since pi also divides n, it divides their difference, n – (n–1) = 1. This is impossible, so there are an infinite number of primes. Results from FactBites:

 Ernst Kummer - definition of Ernst Kummer in Encyclopedia (279 words) Ernst Eduard Kummer (29 January 1810 in Sorau, Brandenburg, Prussia - 14 May 1893 in Berlin, Germany) was a German mathematician. The Kummer surface results from taking the quotient of a two-dimensional abelian variety by the cyclic group {1, -1} (an early orbifold: it has 16 singular points, and its geometry was intensively studied in the nineteenth century). He studied what were later called Kummer extensions of fields: that is, extensions generated by adjoining an n-th root to a field already containing a primitive n-th root of unity.
 Kummer theory - Wikipedia, the free encyclopedia (555 words) In mathematics, Kummer theory provides a description of certain types of field extensions involving the adjunction of nth roots of elements of the base field. The theory was originally developed by Ernst Kummer around the 1840s in his pioneering work on Fermat's Last Theorem. Kummer theory is basic, for example, in class field theory and in general in understanding abelian extensions; it says that in the presence of enough roots of unity, cyclic extensions can be understood in terms of extracting roots.
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