Let p be a prime number, and let s = cos(2 * pi / p) + i * sin(2 * pi / p) be a primitive pth root of unity (ie a complex number such that s^p == 1). Now,

Let p be a prime number, and let s = cos(2 * pi / p) + i * sin(2 * pi / p) be a primitive pth root of unity (ie a complex number such that s^p == 1). Now, consider all complex numbers of the form a_0 + a_1 * s + a_2 * s^2 + ... + a_(p-1) s^(p-1) where the a's (a_0,a_1,...,a_(p-1)) are integers. Note that the fact that s^p == 1 means that we can multiply two things of this form and end up with another thing of this form. Likewise, we can add two things of this form, and get another thing of this form.* For the sake of notation, let's call the set of all such complex numbers Z[s] (where "Z" denotes the set of integers). In 1847, the French mathematician Gabriel Lamé[1] published an attempted proof of Fermat's Last Theorem (ie the assertion that there are no positive integer solutions of x^n + y^n = z^n for n > 2) that carried the implicit assumption that every element of Z[s] except 0, 1, and -1 break down uniquely into irreducibles (much like factoring positive integers: 12 = 2 * 2 * 3, and that's the only way to break 12 down into primes (up to reordering)). However, this uniqueness of factorization doesn't hold when p=23, as discovered by the German mathematician Ernst Kummer[2]. The failure of this approach to proving FLT resulted in the discovery of a significant amount of Algebraic Number Theory, in an attempt to "build more machinery" to get around the problems posed by Lamé's proof. Eventually, Andrew Wiles found a proof of FLT in 1995 that relied on Elliptic Curves and Modular Forms (two fields of mathematics of which I know pretty much nothing). Another, less long-winded example is the Bridges of Königsberg[3]. In 1736, the following problem was posed to Leonhard Euler: is it possible to take a tour of the city of Königsberg in such a way that each bridge is crossed exactly once and you end in the same place where you start? It turns out that the answer to that question is "No", and in his approach to answering the question, Euler laid the foundation for Graph Theory. Also, the roots of Topology can be tied back to this problem, as well. * - In particular, the set of such complex numbers form what's called a ring (http://en.wikipedia.org/wiki/Ring http://en.wikipedia.org/wiki/Ring). [1] http://en.wikipedia.org/wiki/Gabriel_Lam%C3%A9 http://en.wikipedia.org/wiki/Gabriel_Lam%C3%A9 [2] http://en.wikipedia.org/wiki/Ernst_Kummer http://en.wikipedia.org/wiki/Ernst_Kummer [3] http://en.wikipedia.org/wiki/Euler#Graph_theory http://en.wikipedia.org/wiki/Euler#Graph_theory