Suppose we have an irreducible quintic polynomial $f(x)\in \mathbb{Q}[x]$ with 4 complex solutions, say for e.g. $x^5+x^4+x^3-2x^2-2x+5$. It is easy to see that the Galois group $Gal(E/\mathbb{Q})$ has a $5$- cycle.
Now if we label the complex roots as $a_1, a'_1, a_2, a'_2$ where $a'_i$ is the complex conjugate of $a_i$ and look at the automorphisms, $f:a_1 \to a'_1$ (fixing the rest)and $g:a_2\to a'_2$ (fixing the rest). But this does not give any information whether the $Gal(E/\mathbb{Q})\subset S_n$ or $A_n$.
So, what can we do further to get more information about how the Galois group of the polynomial looks like?
Check all the transitive subgroups of $\mathfrak{S}_5$ you will have five groups (up to conjugation).
You have first the subgroup generated by one 5-cycle. Then its normalizer N in $\mathfrak{A}_5$, in $\mathfrak{S}_5$, $\mathfrak{A}_5$ and $\mathfrak{S}_5$.
For instance, if you have a 3-cycle in your Galois group then you are the whole symmetric group or just the antosymmetric one (then compute the discriminant of your polynomial).
To compute Galois group without a computer, I think you should do number theory (that's how you go further in the field).