Galois Groups of Trinomials with Small Coefficients

We calculate the Galois group of $f=x^n+a{x}^k+b\in \mathbb{Z}\left[x\right]$ over $\mathbb{Q}$ for all $1\le \mid ab\mid \le 3$, $n$ sufficiently large and $k$ coprime to $n$. In particular, we prove that the Galois group of $f$ is the maximal one that its factorization type allows. We do it by proving a new sufficient condition for the Galois group of a factor of a trinomial to be symmetric, which perhaps can be used to derive stronger results of this kind in the future.