×

zbMATH — the first resource for mathematics

A class of irreducible polynomials. (English) Zbl 1355.11102
Summary: 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 \[ 3\leq k_{n-1}\leq k_{n-2}\leq \cdots \leq k_1\leq k_0\leq 2k_{n-1}-3. \] 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.
MSC:
11R09 Polynomials (irreducibility, etc.)
11C08 Polynomials in number theory
12D05 Polynomials in real and complex fields: factorization
12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
PDF BibTeX XML Cite
Full Text: DOI