On the Galois group of three classes of trinomials

Let $n\ge 8$ be an integer and let $p$ be a prime number satisfying $\frac{n}{2}<p<n-2$. In this paper, we prove that the Galois groups of the trinomials

$$T_{n,p,k}(x):=x^n+n^k{p}^{(n-1-p)k}{x}^p+n^k{p}^{nk},$$

$$S_{n,p}(x):=x^n+p^{n(n-1-p)}{n}^p{x}^p+n^p{p}^{n^2}$$

and

$$E_{n,p}(x):=x^n+pn{x}^{n-p}+p{n}^2$$

are the full symmetric group $S_n$ under several conditions. This extends the Cohen-Movahhedi-Salinier theorem on the irreducible trinomials $f(x)=x^n+a{x}^s+b$ with integral coefficients.