## 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 .

**Solution:** Say . It’s easy to see that . Therefore, if we consider , then this polynomial satisfies the hypothesis of the following Problem so all the roots of have modulus strictly greater than 1, or has a root which is the root of unity of some order different of 1. Then, since the minimal polynomial of is and is a root of which has integer coefficients, it follows that divides . Suppose are the remainders of modulo , and suppose they are non zero. Then which shows that at least one of is greater or equal to , which is a contradiction, because . Therefore, one of is divisible by . But we know that , so divides the other one too. Therefore , which is a contradiction, because is different from 1 since 1 is not a root for . Therefore has only roots which have modulus greater than 1.

Take a root of . Then is a root for , so . This implies , so has all roots of modulus strictly greater than 1.

Suppose . Then . This implies that one of is 1. Suppose . But . This is a contradiction proving that our assumption was false. Therefore is irreductible.

Hi,

I don’t understand a step in your solution: why is the minimal polynomial of a root of unity is 1+x+x^2+..+x^(k-1)? The minimal polynomial of a root of unity is a cyclotomic polynomial which is not necessarily in the form you wrote. Please help me to clarify what is the problem. :S

And could you mention a book which contains the lemma that helped to solve this problem? I don’t see any other way to find a solution, and I never heard about this lemma before.

Thank you for your help, and keep up the excellent work!

Hello.

I see what you mean. I must modify the solution, using other polynomial as the minimal one.

What lemma are you referring to?

I mean this: https://mathproblems123.wordpress.com/2009/11/09/position-of-roots/

Thanks!

I found the property as a lemma in the solving of the following Romanian National Math Olympiad Test in 1990:

Prove that if and then the polynomial

is irreductible in .

The lemma was stated without the fact that the polynomial can have a root of unity of some order (namely, if the polynomial has coefficients in descending order, and they are not all equal, then all the roots are OUTSIDE the closed unit disk), which I found curious, since the polynomial satisfies the lemma, and for any root of unity of order is a root for the polynomial above.

I don’t know any book which contains the result. I found some posts on mathlinks which use this lemma, but in the “old” form.

Hope this helps.

Beni