 What became clear along such lines is that in some of these extensions the Fundamental Theorem of Arithmetic—asserting the unique factorization of all integers into powers of primes—fails. If it had been available generally, solutions to important problems would have been within reach, including Fermat’s Last Theorem. The question became, then, whether a suitable alternative to the Fundamental Theorem could be found. Kummer’s response, obtained in a close study of certain particular cases, was to introduce “ideal divisors” in terms of which unique factorization could be recovered. While this move led to striking progress, the precise nature of these new mathematical objects was left unclear, as were the basis for their introduction and the range of applicability of the technique.