Find the Galois group of $ f(x) = x^5-6x+3$ over $\mathbb{Q}$.
Let $K$ be the splitting field of $f(x)$. Then $G = Gal(K/\mathbb{Q})$ can be viewed as a group of permutations of the five roots of $f(x)$.
Now $x^5-6x+3$ is irreducible over $\mathbb{Q}$ for Eisenstein's Criterion with $p=3$. This means that $G$ is a transitive subgroup of $S_5$.
I'm not sure where to go now, can someone point me in the right direction?
Best Answer
In general, the Galois group $G$ of an irreducible polynomial $f\in \Bbb Q[X]$ of degree $p$ with $p>2$ prime having exactly $2$ non-real roots is generated by $(12\cdots p)$ and a transposition, which we may assume to be $(12)$, so that $$ G\cong \langle (12\cdots p),(12)\rangle \cong S_p. $$ Here this is satisfied for $p=5$. For further references, see the following duplicates:
Galois group of a degree 5 irreducible polynomial with two complex roots.
Galois group of an irreducible polynomial
f irreducible polynomial with $p-2$ real roots $\Rightarrow$ $Gal(\mathbb{Q}_{f}/\mathbb{Q}) \cong S_{p}$