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Ernst Eduard Kummer (29 January 1810 - 14 May 1893) was a German mathematician. 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). Kummer was first married to Ottilie Mendelssohn, daughter of Nathan Mendelssohn and Henriette Itzig. Ottilie was a cousin of Rebecca Mendelssohn Bartholdy, the wife of the mathematician Peter Gustav Lejeune Dirichlet. His second wife, Bertha was a maternal cousin of Ottilie. Overall, he had 13 children. The daughter Marie married the mathematician Hermann Schwarz. Kummer retired from teaching and from mathematics in 1890 and died three years later in Berlin.

Contributions to mathematics

Kummer made several contributions to mathematics in different areas; he codified some of the relations between different hypergeometric series, known as 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 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.

Kummer surface

Kummer also found the Kummer surface, which is a special case of André Weil's K3 surfaces , possibly 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.

See also

• Kummer series
• Kummer theory

References

• 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.

External links

• O'Connor, John J.; Robertson, Edmund F., "Ernst Kummer", [Expression error: Missing operand for > MacTutor History of Mathematics archive], University of St Andrews, http://www-history.mcs.st-andrews.ac.uk/Biographies/Kummer.html .
• Proof of infinite number of primes
• Biography of Ernst Kummer
• Ernst Kummer at the Mathematics Genealogy Project