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).
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 n-th root to a field already containing a primitive n-th 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.
References
Eric Temple Bell, Men of Mathematics, Simon and Schuster, New York, 1986.
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).
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.
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