Here is a problem from one of the Harvard quals question paper(spring 2011 I think) which I could not figure it out for a long time.
Prove that for any positive integer $a$, the polynomial $f(x)=x^6+3ax^4+3x^3+3ax^2+1$ is irreducible.
I tried to use Eisenstein criterion but I couldn't find a suitable transformation to $f$ so that Eisenstein can be applied and I don't know any other criterion other than this( If we need some other criterion to solve this please mention it).
Please give your ideas/hints to solve this.
Thanks.