TITLE: Galois Group of $x^5+ 5x^3 + 5x + 1$.
QUESTION [7 upvotes]: I've been asked to determine the Galois Group of $x^5+  5x^3 + 5x + 1$. This is what I know so far.
1) The polynomial is irreducible.
2) Its discriminant is $78125=5^7$
Since the discriminant is not a square, the Galois group should be $S_5$ or $F_{20}$. How can I know? In case it was $F_{20}$, is it solvable?
Thanks in advance

REPLY [3 votes]: There is no doubt that this is solvable.  Render $x=r-1/r$ and show that $r^5-1/r^5=-1$.  Thus $r^5=(-1\pm\sqrt5)/2$, etc, and the Galois group can't be larger than $F_{20}$.