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" 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.

Thanks in advance...

 

[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