Help to prove fact about quintic and galois theory?

Question

Consider $f = 2x^5 - 10x +5 \in Q[x]$. Let $L/Q$ be a splitting field of f. Show that $Gal(L/Q)$ injects as a subgroup of $S_5$.

Answer 1

Hint: Show that every automorphism must permute the roots of $f$, and that an automorphism is completely determined by where it sends the roots.