Question

How is Z[√7] a UFD?
UFD means unique factorisation domain. And Z[√7] = {a+b√7: a and b are in Z}, Z is a ring of integers.

Answer 1

Answer:

If nn is even, then 22 divides −n−−−√2=−n−n2=−n but does not divide −n−−−√−n, so 22 is a nonprime irreducible. In a UFD, all irreducibles are prime, so this shows Z[−n−−−√]Z[−n] is not a UFD.

Similarly, if nn is odd, then 22 divides (1+−n−−−√)(1−−n−−−√)=1+n(1+−n)(1−−n)=1+n without dividing either of the factors, so again 22 is a nonprime irreducible.

This argument works equally well for n=3n=3, but fails for n=1,2n=1,2, and in fact Z[−1−−−√]Z[−1] and Z[−2