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Posts Tagged ‘irreductibility’

Irreductible polynomial with paired roots

April 11, 2011 Leave a comment

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.

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Irreductible Polynomials Article AMM

March 26, 2010 Leave a comment

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

January 17, 2010 Leave a comment

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

November 9, 2009 Leave a comment

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.
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