In this article we study the irreducibility of polynomials of the form x n + ǫ 1 x m + p k ǫ 2 , p being a prime number. We will show that they are irreducible for m = 1. We have also provided the cyclotomic factors and reducibility criterion for trinomials of the formThis corrects few of the existing results of W. Ljuggren's on x n + ǫ 1 x m + ǫ 2 .
A cyclotomic polynomial $\unicode[STIX]{x1D6F7}_{k}(x)$ is an essential cyclotomic factor of $f(x)\in \mathbb{Z}[x]$ if $\unicode[STIX]{x1D6F7}_{k}(x)\mid f(x)$ and every prime divisor of $k$ is less than or equal to the number of terms of $f.$ We show that if a monic polynomial with coefficients from $\{-1,0,1\}$ has a cyclotomic factor, then it has an essential cyclotomic factor. We use this result to prove a conjecture posed by Mercer [‘Newman polynomials, reducibility, and roots on the unit circle’, Integers12(4) (2012), 503–519].
In this article, we consider polynomials of the form fWe will show that under the strict inequality these polynomials are irreducible for certain values of n 1 . In the case of equality, apart from its cyclotomic factors, they have exactly one irreducible non-reciprocal factor.
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