Consider $p(x,y)=3x^{5}+3y^{4}x^{4}+6xy+3y+12\in\mathbb{Q}[x,y]$. Is $p(x,y)$ irreducible in $\mathbb{Q}[x,y]$? And in $\mathbb{Z}[x,y]$?

Question

I don't know how to see if there are any $q(x,y),r(x,y)\in\mathbb{Q}[x,y]$ such that $p(x,y)=q(x,y)r(x,y)$ or not. The same in $\mathbb{Z}[x,y]$. I bet it is easier to work in $\mathbb{Q}[x][y]$ and in $\mathbb{Z}[x][y]$, just because the degree of the polynomial becomes lower in such rings. But it does not seem factible to apply any criteria (Eisenstein, Gauss, etc.) Thanks in advance!

Answer 1

HINT:

1. Divide by $3$, get an primitive polynomial, so irreducibility over $\mathbb{Q}$ or $\mathbb{Z}$ is the same ( yes, initially $f$ was reducible over $\mathbb{Z}$, since $3$ not a unit there)

2. Look at it as a polyn in $\mathbb{Z}[y] [x]$. It's monic ( in $x$) so if it were reducible, the factors must have degrees $<5$.

3. Assume you have decomposition in $\mathbb{Z}[y][x]$. Now give the value $y=1$. Get a decomposition for the poly $g(x,1)$ in $\mathbb{Z}[x]$.

4. Check that in fact $g(x,1)$ is irreducible, and get a contradiction. ( work say $\mod 7$ or other way).