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Ernst Kummer
Ernst Eduard Kummer
Born	29 January 1810(1810-01-29) Sorau, Brandenburg, Prussia
Died	14 May 1893 (aged 83) Berlin, Germany
Residence	Germany
Nationality	German
Field	Mathematician
Institutions	University of Berlin University of Breslau Gewerbeinstitut
Alma mater	Universität Halle-Wittenberg
Academic advisor	Heinrich Scherk
Notable students	Georg Frobenius Lazarus Fuchs Hermann Schwarz
Known for	Bessel functions

Ernst Eduard Kummer (29 January 1810 - 14 May 1893) was a German mathematician. Highly skilled in applied mathematics, Kummer trained German army officers in ballistics; afterwards, he taught for 10 years in a Gymnasium (the German equivalent of high school), where he inspired the mathematical career of Leopold Kronecker.

Kummer was born in Sorau, Brandenburg (then part of Prussia). He retired from teaching and from mathematics in 1890 and died three years later in Berlin.

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, Kummer ring and Kummer sum.

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.

Kummer also found the Kummer surface, which is a special case of Andre Weil's K3 surfaces (named after the peak in the Himalayas discovered around the time of Weil's work. Another explanation is that K3 stands for the trio of mathematicians Kummer, Kodaira, and Kähler). K3 surfaces are the Calabi-Yau manifolds of dimension two, and have played an important role in string theory.

• Eric Temple Bell, Men of Mathematics, Simon and Schuster, New York: 1986.

• R. W. H. T. Hudson, Kummer's Quartic Surface, Cambridge, [1905] rept. 1990.

• "Ernst Kummer," in Dictionary of Scientific Biography, ed. C. Gillispie, NY: Scribners 1970-90.

• O'Connor, John J; Edmund F. Robertson "Ernst Kummer". MacTutor History of Mathematics archive.  
• Mathworld, proof of infinite number of primes
• Biography of Ernst Kummer