princeps
Nov 19, 2011, 09:28 AM
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}?
Eisenstein's criterion (http://en.wikipedia.org/wiki/Eisenstein's_criterion) , Cohn's criterion (http://en.wikipedia.org/wiki/Cohn's_irreducibility_criterion) , and Perron's criterion (http://rms.unibuc.ro/bulletin/pdf/53-3/perron.pdf) 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.
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}?
Eisenstein's criterion (http://en.wikipedia.org/wiki/Eisenstein's_criterion) , Cohn's criterion (http://en.wikipedia.org/wiki/Cohn's_irreducibility_criterion) , and Perron's criterion (http://rms.unibuc.ro/bulletin/pdf/53-3/perron.pdf) 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.