Irreducible polynomials over ring of integers ?

Is it true that polynomials of the form :

where , , is odd number , , and

are irreducible over the ring of integers ?

Note that general form of is : , so condition is equivalent to the condition . Also polynomial can be rewritten into form :

Eisenstein's criterion , Cohn's criterion , and Perron's criterion cannot be applied to the polynomials of this form.

Example :

The polynomial is irreducible over the integers but none of the criteria above can be applied on this polynomial.

Re: Irreducible polynomials over ring of integers ?

Quote:

Originally Posted by

**princeps** Is it true that polynomials of the form :

where

,

,

is odd number ,

, and

are irreducible over the ring of integers

?

Note that general form of

is :

, so condition

is equivalent to the condition

. Also polynomial can be rewritten into form :

Eisenstein's criterion ,

Cohn's criterion , and

Perron's criterion cannot be applied to the polynomials of this form.

Example :

The polynomial

is irreducible over the integers but none of the criteria above can be applied on this polynomial.

i don't know the answer to the general case but it's easy to prove that if and then is irreducible.

to see this, note that the numbers and are even and hence

where all are even and thus and so is irreducible by the Eisenstein's criterion.

Re: Irreducible polynomials over ring of integers ?

Quote:

Originally Posted by

**NonCommAlg**
where all

are even and

There is condition in the text of the question , where is an odd number so cannot be even number.

Re: Irreducible polynomials over ring of integers ?

Quote:

Originally Posted by

**princeps** There is condition in the text of the question

, where

is an odd number so

cannot be even number.

in my solution, are the coefficients of not .

Re: Irreducible polynomials over ring of integers ?

Can you give me an example of f(x+1) ?