Algebraic Number Theory and Hamiltonian Chaos
Three great themes were developed in algebraic number theory during the nineteenth century, namely quadratic forms, algebraic numbers and ideals, which call to mind the names of Gauss, Kummer, and Dedekind, respectively. They represent three ways of dealing with one and the same problem, the failure of the fundamental theorem of arithmetic (unique factorization into primes) in algebraic number fields. Had the fundamental theorem been valid, Gauss would have probably answered any conceivable question about binary quadratic forms while in his teens, and Kummer would have proved Fermat’s last theorem. As to ideal theory, it wouldn’t have been developed for quite a while.
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