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
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.
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:
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
is a primitive p^th root of unity, then the ring of algebraic 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
[2]: http://people.math.gatech.edu/~jrabinoff6/mathcamp/lectures....
[3]: http://en.wikipedia.org/wiki/Regular_prime