Compute the splitting field of $X^4 + 5 X^3 + 10 X^2 + 10 X + 5$ over $\Bbb Q$

Question

I'm trying compute the splitting field of $P(X) := X^4 + 5 X^3 + 10 X^2 + 10 X + 5$ over $\mathbb{Q}$.

This is what I thought: I tried find the roots of $P$ observing that

$$P(X-1) = X^4 + X^3 + X^2 + X + 11$$

but I'm stuck here.

Can anyone give me a direction in order to find the splitting field of $P(X)$?

Answer 1

Finding roots of this polynomial is difficult so we use the following result:

Result: Let $f (x) \in F[x]$ and let $a \in F$. Then $f(x)$ and $f(x+ a)$ have the same splitting field over $F$.

Here $-1 \in \Bbb{Q}$ and $f(x+(-1))=1+x+x^2+x^3+x^4$ not $11$ in the constant term

But the last polynomial is a Cyclotomic polynomial. So comparing to original polynomial, this one is easy to find!