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Eduard Kummer's main achievement was the extension of results about the integers to other integral domains by introducing the concept of an ideal.
Kummer taught for one year (1833) at Sorau and then for ten years at Liegnitz. In 1855 Dirichlet's chair at the University became vacant and Kummer was appointed.
In 1843 Kummer, realising that attempts to prove Fermat's Last Theorem broke down because the unique factorisation of integers did not extend to other rings of complex numbers, attempted to restore the uniqueness of factorisation by introducing 'ideal' numbers. Not only has his work been most fundamental in work relating to Fermat's Last Theorem, since all later work has been based on it, but the concept of an ideal allowed ring theory, and much of abstract algebra, to develop.
The Paris Academy of Sciences awarded him the Grand Prize in 1857 for this work. In fact the prize of 3000 francs was offered for a solution to Fermat's Last Theorem but when no solution was forthcoming, even after extending the date, the Prize was given to Kummer even though he had not submitted an entry for the Prize.
Kummer also studied the surface, now named after him, based on the singular surface of the quadratic line complex.
He also worked on extending Gauss's work on hypergeometric series, giving developments that are useful in the theory of differential equations.
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Tell me about Kummer's work on Fermat's last theorem
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