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 p-cyclotomic integers

Z[] = {a₀ + a₁ + 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 re-examined 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) = a² + 5b²

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

= a²c² - 10abcd + 25b²d² + 5b²c² + 10abcd + 5a²d²

= a²c² + 5b²c² + 5a²d² + 25b²d²

= (a² + 5b²)(c² + 5d²)

= 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 a² + 5b² = 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.