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!
Best Answer
It looks like there may really have been a 9-piece McNuggets in the 80’s, but given the circumstances I’m inclined to think someone sloppily copied a 4 into a 9. That would fix all your problems.
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.