On the Galois group of three classes of trinomials

Abstract

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^kp^{(n-1-p)k}x^p+n^kp^{nk}, $ $ S_{n, p}(x): = x^n+p^{n(n-1-p)}n^px^p+n^pp^{n^2} $ and $ E_{n, p}(x): = x^n+pnx^{n-p}+pn^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+ax^s+b $ with integral coefficients.

Keywords

• p-adic newton polygon
• irreducibility criterion
• galois group