Question

Give an example to show that a subdomain of a unique factorization domain need not be a UFD.

Short answer

It is shown that a subdomain of a unique factorization domain need not be a UFD.

Step-by-step solution

Definition of UFD

Unique Factorization Domain (UFD)

It is an integral domain in which each non-zero and non-invertible element has a unique factorization.

Proving that a subdomain of a unique factorization domain need not be a UFD

Consider the field of complex numbers $\mathrm{ℂ}$.

As we know every field is a UFD, therefore, $\mathrm{ℂ}$is also UFD.

Let’s consider subdomain $\mathbb{Z}[-\sqrt{5}]\in ℂ$.

Now, considering element 6 in $\mathrm{\mathbb{Z}}[-\sqrt{5}]$.

We have, $6=2\times 3$and$6=(1+\sqrt{-5})\times (1-\sqrt{-5})$ .

Now, we must show that 2 and 3 are irreducible in$\mathbb{Z}[-\sqrt{5}]$ .

Let’s assume, $2=(p+q\sqrt{-5})(r+s\sqrt{-5})$for some integers p, q, r, and s.

This implies that, either $(p+q\sqrt{-5})$is a unit or$(r+s\sqrt{-5})$ is a unit.

Proving 2 as an irreducible element in$\mathbb{Z}[-\sqrt{5}]$ .

This also implies that $(1+\sqrt{-5})$and $(1-\sqrt{-5})$are irreducible in $\mathbb{Z}[-\sqrt{5}]$.

Thus, we can conclude that there are two different factorizations of 6 into irreducible, and none of the factors 2 and 3 are associates of$(1+\sqrt{-5})$ or $(1-\sqrt{-5})$.

So, 1 and -1 are the only unit in$\mathbb{Z}[-\sqrt{5}]$ .

This implies$\mathbb{Z}[-\sqrt{5}]$ is not UFD.

Hence, it is proved that a subdomain of a unique factorization domain need not to be a UFD.