# Show that the McNuggets ring $A=\mathbb C[z^6,z^9,z^{20}]$ is not a UFD

Let $$A=\mathbb C[z^6,z^9,z^{20}]$$ be the McNuggets ring. Is $$A$$ a unique factorisation domain?

I'm reading a solution to this question and it says,

No, since $$z^{12}=z^6z^6=z^4z^4z^4$$ which are two distinct factorisations so $$A$$ is not a UFD.

Firstly I want to ask is this ring essentially the ring of polynomials with $$z^6,z^9,z^{20}$$ as variables with coefficients in $$\mathbb C$$? If so then how can $$z^4\in A$$?

And secondly why does $$z^{12}=z^6z^6=z^4z^4z^4$$ imply it's not a UFD? Is it because $$z^6$$ and $$z^4$$ are irreducible in $$A$$? If so why? Thanks!

• It sounds like a typo / thinko in the book, but the result still holds — consider $z^{18} = z^9\times z^9 = z^6\times z^6\times z^6$. Commented May 10, 2020 at 20:50

Notice that a factorization in this ring would also be a factorization in $$\mathbb C[z]$$, so that limits factorizations to powers of $$z$$.
$$z^4$$ is clearly irreducible, since it is the lowest positive power of $$z$$ you can produce.
$$z^6$$ is higher, but the only candidate for a factor is $$z^4$$, and $$z^2$$ isn't available. So it's also irreducible.