Let $f(x)=x^5+10x^3+10x+3$ be a polynomial in $\Bbb{Z}[i][x]$. We have to check whether it is irreducible in $\Bbb{Q}(i)[x]$ or not.
As $\Bbb{Q}(i)$ is quotient field of $\Bbb{Z}[i]$ and $f$ is primitive, $f(x)$ is irreducible in $\Bbb{Z}[i][x]$ iff $f(x)$ is irreducible in $\Bbb{Q}(i)[x]$.
I am not able to apply Eisenstein Criterion here. Is there any way out to solve this kind of problems without explicitly finding the roots?
Can anyone give me any idea how to solve the problem? Thanks for help in advance.