For $x^2+x+1$, there exist infinitely many integers $w$ such that the splitting field of $x^2+x+1+w$ is the same field as $x^2+x+1$. Examples of such integers $w$ are $$6, 60, 546, 4920,\ldots\,.$$ There is a closed form for odd $k$, namely $$w=\frac{3^k+1}{4}-1\,.$$
I can't seem to prove / disprove this for the fact of whether there are infinitely many integers $w$ such that the splitting field of $x^4+x^3+x^2+x+1+w$ is the same field as the splitting field $x^4+x^3+x^2+x+1$.
The same with whether there are infinitely many integers $w$ such that the splitting field of $x^6+x^5+x^4+x^3+x^2+x+1+w$ is the same field as the splitting field $x^6+x^5+x^4+x^3+x^2+x+1$.
If there is at least one integer $w$ for degrees $4$ or $6$, please post them as a counterexample.