# Irreducible polynomials over ring of integers

pedja
Is it true that polynomials of the form :

$f_n= x^n+x^{n-1}+\cdots+x^{k+1}+ax^k+ax^{k-1}+\cdots+a$

where $\gcd(n+1,k+1)=1$ , $a\in \mathbb{Z^{+}}$ , $a$ is odd number , $a>1$, and $a_1\neq 1$

are irreducible over the ring of integers $\mathbb{Z}$?

http://en.wikipedia.org/wiki/Eisenstein%27s_criterion" [Broken] cannot be applied to the polynomials of this form.

Example :

The polynomial $x^4+x^3+x^2+3x+3$ is irreducible over the integers but none of the criteria above can be applied on this polynomial.

Last edited by a moderator:

COURAGE_WOLF
I hope this helps.

The polynomial x4+x3+x2+x +1 is the cyclotomic polynomial $\Phi$5(x). I believe the general polynomial that you described is also the monic cyclotomic polynomial $\Phi$n(x) in Z[x] with degree $\varphi$(n). These polynomials are irreducible. The proof is a little tedious and not exactly immediate. Check out this paper

http://www.math.umn.edu/~garrett/m/algebra/notes/08.pdf

I think it does a pretty good job. Then again, if that general polynomial turns out to not be cyclotomic, I'll have to start over and come up with something new

pedja
These polynomials are not cyclotomic polynomials.
$f_n$ can be rewritten into form :

$f_n=\displaystyle \sum_{i=0}^n x^{i}+(a-1)\cdot \displaystyle \sum_{i=0}^k x^{i}$ ,or

$f_n=\frac{x^{n+1}+(a-1)x^{k+1}-a}{x-1}$

COURAGE_WOLF
Maybe if you give a different example it would be easier to see. I don't really see much of a correlation between the general case and the example you gave. Anyways, you can always fall back on the general technique for finding irreduciblity.

Let I be a proper ideal in the integral domain R and let p(x) be the monic polynomial in R[x]. If the image of p(x) in (R/I)[x] cannot be factored in (R/I)[x] into two polynomials of smaller dgree, then p(x) is irreducible in R[x].

Start reducing mod some n and see where that gets you. If it's small degree then it's pretty obvious. A higher degree will probably take a little work. Unfortunately, this technique doesn't always work, but I can't think of any other irreducibility criteria.

COURAGE_WOLF
Wait I just thought of another one: Hilbert's Irreducibility Theorem.

COURAGE_WOLF
Also Kronecker's Method