Determine Galois groups of polynomials Determine the Galois groups of the following polynomials over Q.
$f(x)=x^3−3x+1$.
$g(x)=x^4+3x+3$.
$h(x)=x^5+8x+12$. 
I have no way to find their Galois groups. I only obtained that since they are irreducible, their Galois groups are transitive subgroups of $S_3$, $S_4$, $S_5$ resp. Should I try all possibilities one by one?

Well, for the first one it is is $S_3$ or $A_3.$ The latter possibility only arises if the discriminant is a perfect square, so you should compute that. For the second polynomial, you need to check whether the discriminant is a square AND whether the cubic resolvent is irreducible (the cubic resolvent for $g$ is $x^3 - 12x - 9$). In fact, the discriminant is the same for the quartic and the cubic resolvent. If the cubic resolvent is irreducible the Galois group is either $S_4$ or $A_4,$ depending on whether the discriminant is a square. For the last polynomial, you should factor it modulo some primes and see what happens. For example, mod $3,$ the polynomial factors as $x(x^2+1)(x+1)(x-1),$ so your Galois group has a transposition, which, presumably, means that it is all of $S_5.$ 
