Z[] is not a UFD
In 1835 Ernst Kummer believed that he had proved Fermat's last theorem. His proof relied on the assumption that the rings of pcyclotomic integers
Z[ ] = {a _{0} + a _{1} + a _{r}^{r}  a _{i} Z, r > 0}.
were unique factorization domains (UFD's). Dirichlet pointed out to Kummer that a similar, but simpler, ring Z[] was not a UFD. Kummer reexamined the rings Z[] and found that they were not UFD's. Let us take some time to examine Z[] and why it is not a unique factorization domain.
Z[] is defined to be the set of complex numbers Z[] = {a + b  a,b Z}. and over this set addition is defined by
(a + b ) + (c + d ) = (a + c) + (b + d) ,
and multiplication by
(a + b ) · (c + d ) = (ac  5bd) + (bc + ad) .
It is a very simple mater to show that Z[] with these two operations defines a ring. Let us als define the norm mapping
N : Z[ ] Z
by
N(a + b ) = a ^{2} + 5b ^{2}
If we set x = a + b and y = c + d. Then by the above definitions we see that
N(xy) = N((ac  5bd) + (bc + ad))
= (ac 5bd)^{2} + 5(bc + ad)^{2}
= a^{2}c^{2}  10abcd + 25b^{2}d^{2} + 5b^{2}c^{2} + 10abcd + 5a^{2}d^{2}
= a^{2}c^{2} + 5b^{2}c^{2} + 5a^{2}d^{2} + 25b^{2}d^{2}
= (a^{2} + 5b^{2})(c^{2} + 5d^{2})
= N(x)N(y)
Now let us consider the case N(3) = 9, if x is a divisor of 3, N(x) is a divisor of 9, so that N(x) = 9, 3 or 1. In the first case, x = ±3. In the second case x cannot exist since a^{2} + 5b^{2} = 3 has no integer solutions. Similarly N(2) = 9
and N(2 + ) = 9 both of which are also prime in Z[]. But 3 · 3 = (2  )·(2 + ) = 9. So an element of Z[] can be represented by different combinations of primes, hence is not a unique factorization domain.
