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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)
##### Keywords:
irreducible polynomial; Eneström-Kakeya
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