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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