Yes, the article did a vast oversimplification there. In particular, Cauchy, Lamé, and Kummer were working with rings of algebraic integers[1]. One simple exa

Yes, the article did a vast oversimplification there. In particular, Cauchy, Lamé, and Kummer were working with rings of algebraic integers[1]. One simple example is the ring denoted Z[sqrt(-5)], which is the set of all complex numbers whose real part is an integer and whose imaginary part is an integer multiple of sqrt(5), ie Z[sqrt(-5)] = { a + b * sqrt(-5) | a,b are integers } This ring has a lot of structure in common with the integers, but it does not have unique factorization. In particular: 6 = 2 * 3 = (1 + sqrt(-5)) * (1 - sqrt(-5)) and each of 2, 3, 1 + sqrt(-5) and 1 - sqrt(-5) are irreducible in Z[sqrt(-5)] (ie they don't have any other factors other than 1 or -1). This is in opposition to what happens in the integers, namely unique factorization of any integer (other than 0, 1, or -1) into primes. The failed proof mentioned above hinged on the following assumption: if p is a prime and r_p = cos( 2 * pi / p) + i * sin( 2 * pi / p) is a primitive p^th root of unity, then the ring of algebraic integers Z[r_p] = {a_0 + a_1 * r_p + a_2 * (r_p)^2 + ... + a_(p-1) * (r_p)^(p-1) | a_0, a_1, ... , a_(p-1) are integers} has unique factorization[2]. It turns out that Z[r_p] has unique factorization if p is a regular prime[3]. However, this is not the case for non-regular primes. Thus, Fermat's Last Theorem wasn't settled in the 19th century. On the bright side, however, these failed attempts contributed to renewed interest in algebraic number theory. [1]: http://en.wikipedia.org/wiki/Algebraic_integer http://en.wikipedia.org/wiki/Algebraic_integer [2]: http://people.math.gatech.edu/~jrabinoff6/mathcamp/lectures.pdf http://people.math.gatech.edu/~jrabinoff6/mathcamp/lectures.... [3]: http://en.wikipedia.org/wiki/Regular_prime http://en.wikipedia.org/wiki/Regular_prime