My Question is to check Irreducibility for polynomials not satisfying Eisenstein Criterion.
As an Illustration, to check whether $x^{p-1}+.....+x+1$ for p a prime is irreducible or not, we replaced $x$ by $x+1$ and by using Eisenstein's Criterion for the resulting polynomial we conclude that resulting polynomial is irreducible and so is the original polynomial.
In general, for a given irreducible polynomial $f(x)$ with coefficients in a known U.F.D, is there some element $a$ such that we can apply Eisenstein's criterion to $f(x+a)$?
I am sure there would be no general structure for this but I expect there to be at least some special cases.
Any Reference/suggestion would be appreciated.
Thank You.
Do Eisenstein and linear translations suffice to determine irreducibility of any integer polynomial? The answer is no. Given $f(x)=a_nx^n+a_{n-1}x^{n-1}+\ldots +a_0$ there is little choice in the linear translation to be applied. In the special case $a_n=1$, $a_{n-1}=0$, the coefficient of $x^{n-1}$ in $f(x+a)$ is $na$, so we must take $a\equiv 0\pmod p$, i.e. we stay with the given $f$, unless $p|n$. On th eother hand, if $p|n$ then the coefficient of $x^{n-2}$ becomes $a_{n-2}+{p\choose 2}a$ and this is $\equiv a_{n-2}\pmod p$ unless $p=2$. Thus if we exhibit any irreducible polynomial with $n$ odd, $a_n=1$, $a_{n-1}=0$ and such that Eisenstein cannot show its irreducibility, then neither Eisenstein plus linear translations can show irreducibility. One such polynomial is $$f(x)=x^3+x+1\in\mathbb Z[x].$$