From: rusin@washington.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: why can't you factorize this polynomial?
Date: 26 Oct 1995 20:43:55 GMT

In article <stolting.814560929@leibniz.math.fu-berlin.de>,
Gregor Stoelting <stolting@math.fu-berlin.de> wrote:
>E(n,k):= n^2 + k^2 +1
>
>P(n,k):=  
>
> l               j-1              l
>----             ----            ----
>/      sigma_j(n) ||   E(n-i, k)  ||   E(n-i, k-1)
>\                 ||              ||
>----             ----            ----
>j= 0             i=0             i=j
>
>where sigma_j are nonzero constants according to k. 
>Is it possible to proof, that this polymomial
>doen't have nontrivial factors in Q[k]?

Well, it has the factor  E(n-j,k-1). Perhaps you meant for the last
product to be taken over   i=j,...,l-1 ?

Even then it seems you must know something about your  sigma_j.
What if all sigma_j(n)=0  except for one of them? Then  P  is clearly
a product. A less trivial example occurs with  l=1, sigma_0=1,
sigma_1=-2/3. Then  P(-1,k)=E(1,k-1)+(-2/3)E(1,k)= (1/3)(k-1)(k-5).

dave