Let $x^4+x^3y+x^2y^2+xy^3+y^4 \in \mathbb{Q}[x,y]$ be a primitive polynomial for which we have to investigate if it is irreducible. My idea is:
since $(y-1) \in \mathbb{Q}[x,y]$ is a prime ideal such that $\overline{x^4+x^3y+x^2y^2+xy^3+y^4}=\overline{x^4+x^3+x^2+x+1}$ in $(\mathbb{Q}[y] / (y-1))[x] \cong \mathbb{Q}[x]$ and $x^4+x^3+x^2+x+1$ is irreducible in $\mathbb{Q}[x]$, then the first polynomial is irreducible. Is this reasoning correct?
The reasoning is correct. Let's suppose that $$x^4+x^3y+x^2y^2+xy^3+y^4=f(x,y)g(x,y)$$ in $\mathbb Q[x,y]$. Clearly $\deg_xf\ge 1$ and $\deg_xg\ge 1$. Now use the Reduction criterion for irreducibility: let $A=\mathbb Q[y]$, $B=\mathbb Q$, and $\sigma:A\to B$, $\sigma(y)=1$.