Does there exist an algorithm which computes the Galois group of a polynomial 
$p(x) \in \mathbb{Z}[x]$? Feel free to interpret this question in any reasonable manner. For example, if the degree of $p(x)$ is $n$, then the algorithm could give a set of permutations $\pi \in Sym(n)$ which generate the Galois group.



Computing the Galois group of a polynomial
There is an algorithm described in an ancient and interesting book on Galois Theory by Leonard Eugene Dickson. Here is a brief sketch in the case of an irreducible polynomial $f\in \mathbb{Q}[x]$.


Suppose that $z\_1\ldots z\_n$ are the roots of $f$ in some splitting field of $f$ over $\mathbb{Q}$. (We don't need to construct the splitting field. The $z\_i$ are mentioned here for the sake of explanation.) Let $x\_1\ldots x\_n$ be indeterminates. For a permutation $\sigma\in S\_n$, let
$$E\_\sigma=x\_1z\_{\sigma(1)}+\ldots+ x\_n z\_{\sigma(n)}.$$
Let $g(x):=\prod \_{\sigma} (x-E\_\sigma)$,
where $\sigma$ runs through all permutations in $S\_n$. Each coefficient $c\_i$ of $x^i$ in $g$ is symmetric in $z\_1 \ldots z\_n$, so (using the theorem on symmetric functions) we can write $c\_i$ as a polynomial in $x\_1\dots x\_n$ with rational coefficients.


Assuming that this has been done, factor $g$ into irreducibles over the ring $\mathbb{Q}[x\_1 \ldots x\_n]$. Let $g\_0$ be the irreducible factor of $g$ that is satisfied by $E\_{Id}$, where $Id$ is the identity permuation. Then the galois group of $f$ consists of all permutations of $x\_1\ldots x\_n$ that fix $g\_0$.


The point is that the computation of $g\_0$ is effective (albeit horrendous) and so is the determination of the permutations that fix $g\_0$.


