How to show that the polynomial $ab^3 + cd^3$ is irreducible?

Question

I have to show that the polynomial:

$$ab^3 + cd^3 \in \mathbb{C}[a,b,c,d]$$

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.

Answer 1

If $ab^3 + cd^3=f(a,b,c,d)g(a,b,c,d)$, then $\deg_af+\deg_ag=1$. Suppose $\deg_af=0$, and $\deg_ag=1$, so $f\in\mathbb C[b,c,d]$ and $g=h(b,c,d)a+k(b,c,d)$. We get $$ab^3 + cd^3=f(b,c,d)[h(b,c,d)a+k(b,c,d)].$$ Then $f(b,c,d)h(b,c,d)=b^3$ and $f(b,c,d)k(b,c,d)=cd^3$. What do we get from here?

Answer 2

This question might also be a good excuse to mention Gauss' lemma and Eisenstein's criterion for irreducibility. Most often these seem to be mentioned just for $\mathbb Z[x]$ (and $\mathbb Q[x]$), but they do apply to $\mathbb C[x,y,z,w]$ and other unique factorization domains, as well.

This puts a slightly different emphasis on the direct argument give by @user26857. Yes, based on degree considerations, a linear polynomial in $x$, with coefficients in the field of rational functions $\mathbb C(y,z,w)$, is irreducible in the PID $\mathbb C(y,z,w)[x]$. With a little more detail about those coefficients, Gauss' lemma gives the irreducibility in $\mathbb C[x,y,z,w]$.

Gauss' lemma and Eisenstein's criterion do also apply to algebraic examples wherein explicit formulaic discussion becomes much messier. (Indeed, those two results nicely package a useful family of computational ideas.)