To get a better feeling for Galois groups I'd like to know some general cases that allow to tell a Galois group from a polynomial and vice versa.
The most simple example I came about is
$P(x)\in \mathbb{Q}[x]$ splits over $\mathbb{Q}$ iff its Galois group is trivial.
I'm looking for other "simple" properties $\mathcal{P}$ of polynomials and properties $\mathcal{G}$ of groups for which it holds:
$P(x)$ is $\mathcal{P}$ iff its Galois group is $\mathcal{G}$.
Especially:
For which polynomials is the Galois group a full symmetric group?
For which polynomials is the Galois group an alternating group?
For which polynomials is the Galois group a cyclic group?
Which general property do the Galois groups of the polynomials $X^n - 1$ have?
Which general property do the Galois groups of cyclotomic polynomials have?