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Is it true that polynomials of the form :

[itex] f_n= x^n+x^{n-1}+\cdots+x^{k+1}+ax^k+ax^{k-1}+\cdots+a[/itex]

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

are irreducible over the ring of integers [itex]\mathbb{Z}[/itex]?

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

Example :

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

Thanks in advance...

[itex] f_n= x^n+x^{n-1}+\cdots+x^{k+1}+ax^k+ax^{k-1}+\cdots+a[/itex]

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

are irreducible over the ring of integers [itex]\mathbb{Z}[/itex]?

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

Example :

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

Thanks in advance...

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