Make your own free website on Tripod.com

Z[square root of -5] 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[zeta] = {a0 + a1zeta + arzetar | ai element of Z, r > 0}.

were unique factorization domains (UFD's). Dirichlet pointed out to Kummer that a similar, but simpler, ring Z[square root of -5] was not a UFD. Kummer re-examined the rings Z[zeta] and found that they were not UFD's. Let us take some time to examine Z[square root of -5] and why it is not a unique factorization domain.

Z[square root of -5] is defined to be the set of complex numbers Z[square root of -5] = {a + bsquare root of -5 | a,b element of Z}. and over this set addition is defined by

(a + bsquare root of -5) + (c + dsquare root of -5) = (a + c) + (b + d)square root of -5,

and multiplication by

(a + bsquare root of -5) · (c + dsquare root of -5) = (ac - 5bd) + (bc + ad)square root of -5.

It is a very simple mater to show that Z[square root of -5] with these two operations defines a ring. Let us als define the norm mapping

N : Z[square root of -5]mapsZ

by

N(a + bsquare root of -5) = a2 + 5b2

If we set x = a + bsquare root of -5 and y = c + dsquare root of -5. Then by the above definitions we see that

N(xy) = N((ac - 5bd) + (bc + ad)square root of -5)

= (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-square root of -5) = 9 and N(2 + square root of -5) = 9 both of which are also prime in Z[square root of -5]. But 3 · 3 = (2 - square root of -5)·(2 + square root of -5) = 9. So an element of Z[square root of -5] can be represented by different combinations of primes, hence is not a unique factorization domain.