Galois Groups of Trinomials with Small Coefficients

Source abstract

We calculate the Galois group of $f = x^n + ax^k + b \in \mathbb{Z}\left[ x \right]$ over $\mathbb{Q}$ for all $1 \leq \left| ab \right| \leq 3$, $n$ 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.