## Irreductible polynomial with paired roots

Suppose is an irreductible polynomial which has a complex root such that is also a root for . Prove that for any other root of , is also a root for .

## Irreductible Polynomial TST 2003

Let be an irreducible polynomial over the ring of integer polynomials, such that is not a perfect square. Prove that if the leading coefficient of is 1 (the coefficient of the term having the highest degree in ) then is also irreducible in the ring of integer polynomials.

*Mihai Piticari, Romanian TST 2003*

## Irreductible polynomial

Suppose is a polynomial with integer coefficients such that is an odd prime, and the roots of form a regular polygon in the complex plane. Prove that is irreductible in .

*Ioan Baetu, Romanian Mathematical Gazette*

## Irreductible polynomials

1) Prove that is irreductible over , where are pairwise different.

2) Prove that is irreductible over , where are pairwise different with one single exception, namely, .

3) Prove that is irreductible over , where are pairwise different.

4) Prove that if a polynomial of degree has integer coefficients and takes the values for more than then this polynomial is irreductible. ( is the greater integer not exceeding )

## Irreductible polynomial 1

Prove that if and then the polynomial is irreductible in .

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## Switch the coefficients

Let a polynomial of degree having integer coefficients; then is irreductible in if and only if is irreductible in .

**Solution:** Suppose is irreductible and , with . Then when are two polynomials of degree at least 1. Therefore is irreductible.

## Irreductible Polynomial

Prove that the polynomial is irreductible over for all integers .

*Miklos Schweitzer 2009 Problem 4*

A generalization of this problem was proposed by myself in the Romanian TST 2010, the solution being similar to the one below. The generalization said to prove that if is a prime and are positive integers with then is irreductible over .

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