I have to show that the polynomial:
$$
a b^{3}+c d^{3} \in \mathbb{C}[a, b, c, d]
$$
cannot be factorised into polynomials of lower degrees, i.e. it is not reducible. However, I’m quite unsure on how to proceed here. I thought I could try to factorise this into a linear times a cubic term and reach a contradiction but involves dealing with dozens of terms and I don’t think it’s the best strategy.
If $a b^{3}+c d^{3}=f(a, b, c, d) g(a, b, c, d),$ then $\operatorname{deg}_{a} f+\operatorname{deg}_{a} g=1 .$ Suppose $\operatorname{deg}_{a} f=0,$ and
$\operatorname{deg}_{a} g=1$, so $f \in \mathbb{C}[b, c, d]$ and $g=h(b, c, d) a+k(b, c, d) .$ We get
$$
a b^{3}+c d^{3}=f(b, c, d)[h(b, c, d) a+k(b, c, d)]
$$
Then $f(b, c, d) h(b, c, d)=b^{3}$ and $f(b, c, d) k(b, c, d)=c d^{3}$. What do we get from here?