A class of irreducible polynomials
Volume 132 / 2013
Colloquium Mathematicum 132 (2013), 113-119
MSC: Primary 11R09; Secondary 11C08.
DOI: 10.4064/cm132-1-9
Abstract
Let \[ f(x)=x^n+k_{n-1}x^{n-1}+k_{n-2}x^{n-2}+\cdots +k_1x+k_0\in \mathbb {Z}[x], \] where \[ 3\le k_{n-1}\le k_{n-2}\le \cdots \le k_1\le k_0\le 2k_{n-1}-3. \] We show that $f(x)$ and $f(x^2)$ are irreducible over $\mathbb {Q}$. Moreover, the upper bound of $2k_{n-1}-3$ on the coefficients of $f(x)$ is the best possible in this situation.
