Following a discussion with a friend of mine, we would like to ask:
Is $xy$ considered irreducible in $\mathbb{C}[x,y]$?
It admits a factorisation $xy=(x)(y)$ where $x,y \in \mathbb{C}[x,y]$ so I am inclined to say no, but my friend thinks otherwise.
Thanks in advance!
A very stupid question, I know - I apologise!
$xy$ is certainly not irreducible, it's a product of $x,y \in \mathbb{C}[x,y]$, which are both non-units of $\mathbb{C}[x,y]$ (exercise). So it's reducible by definition and so by simple 2-valued logic "not irreducible.