### Archive

Posts Tagged ‘irreductibility’

## Irreductible polynomial with paired roots

Suppose $f \in \Bbb{Q}[X]$ is an irreductible polynomial which has a complex root $a$ such that $-a$ is also a root for $f$. Prove that for any other root $b$ of $f$, $-b$ is also a root for $f$.

## Irreductible Polynomials Article AMM

Let $f(x)=a_mx^m+a_{m-1}x^{m-1}+...+a_1x+a_0$ be a polynomial of degree $m$ in $\mathbb{Z}[X]$ and set $\displaystyle H= \max_{0\leq i \leq m-1} \left |\frac{a_i}{a_m}\right|$.
If $f(n)$ is prime for some integer $n \geq H+2$, then $f$ is irreductible in $\mathbb{Z}[X]$. Read more…

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## Irreductible polynomial

Suppose $f=X^p+a_{p-1}X^{p-1}+...a_1X+p$ is a polynomial with integer coefficients such that $p$ is an odd prime, and the roots of $f$ form a regular polygon in the complex plane. Prove that $f$ is irreductible in $\mathbb{Q}[X]$.
Ioan Baetu, Romanian Mathematical Gazette

## Position of roots

The following result is quite useful in irreductibility problems for polynomials.

Prove that if $a_0 \geq a_1\geq ... \geq a_n >0$ then all the roots $z$ of $p(x)=a_nx^n+...+a_1x+a_0$ satisfy $|z| >1$ or $z$ a root of unity of some order.