is the polynomial $\ x^3 - y^5$ irreducible in $\ \Bbb Q[x, y]$

Question

Is the polynomial $\ x^3 - y^5$ irreducible in $\ \Bbb Q[x, y]$.

I know that we need to think of the polynomial as $\ \Bbb Q[x] [y]$. But all examples I have seen involve a $\ y^2$ and we have $\ y^5$... Can I still use the method of taking $\ [()+()][ℎ()+()]$ and expand the brackets out? Any hints or comments would be great thank you.

Answer 1

You can consider $f(x,y)=x^3-y^5$ as polynomial $f(x)\in\mathbb{Q}(y)[x]$. For a degree 3 polynomial it is enough to test for a root. Let's assume there is an element $r=\frac{p(y)}{q(y)}$ with $f(r)=0$ which leads to a contradiction.

Answer 2

If you had a factorization $f(x,y) = g(x,y)h(x,y)$, then for a fixed value of $y$, for example say $y = 2$, you have a factorization $f(x,2) = g(x,2)h(x,2)$. So you have $$x^3 - 32 = g(x,2)h(x,2)$$ But you can readily show $x^3- 32$ is irreducible over ${\mathbb Q}$, leading to that either $g(x,2)$ or $h(x,2)$ is constant. If you do this for enough values of $y$, this will imply that $g(x,y)$ or $h(x,y)$ is a constant polynomial, leading to the conclusion that $f(x,y)$ is irreducible.