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Math Help - Irreducible polynomials over ring of integers ?

  1. #1
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    Irreducible polynomials over ring of integers ?

    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}?

    Note that general form of  f_n is :  f_n=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0 , so condition a_1 \neq 1 is equivalent to the condition  k \geq 1 . Also polynomial can be rewritten into form :

     f_n=\frac{x^{n+1}+(a-1)x^{k+1}-a}{x-1}

    Eisenstein's criterion , Cohn's criterion , and Perron's 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.
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  2. #2
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    Re: Irreducible polynomials over ring of integers ?

    Quote Originally Posted by princeps View Post
    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}?

    Note that general form of  f_n is :  f_n=a_nx^n+a_{n-1}x^{n-1}+\cdots+a_1x+a_0 , so condition a_1 \neq 1 is equivalent to the condition  k \geq 1 . Also polynomial can be rewritten into form :

     f_n=\frac{x^{n+1}+(a-1)x^{k+1}-a}{x-1}

    Eisenstein's criterion , Cohn's criterion , and Perron's 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.
    i don't know the answer to the general case but it's easy to prove that if n = 2^m - 1, \ m \geq 2, \ 2 \mid k and a \equiv 3 \mod 4, then f is irreducible.
    to see this, note that the numbers \binom{2^m}{i}, \ 0 < i < 2^m, and a - 1 are even and hence

    f(x+1) = \frac{(x+1)^{n+1}+(a-1)(x+1)^{k+1}-a}{x}=x^{n} + a_{n-1}x^{n-1}+ \ldots + a_1x + a_0,

    where all a_j are even and a_0 = 2^m + (k+1)(a-1) \not \equiv 0 \mod 4. thus f(x+1), and so f(x), is irreducible by the Eisenstein's criterion.
    Last edited by NonCommAlg; November 27th 2011 at 08:32 PM.
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    Re: Irreducible polynomials over ring of integers ?

    Quote Originally Posted by NonCommAlg View Post
    f(x+1) = \frac{(x+1)^{n+1}+(a-1)(x+1)^{k+1}-a}{x}=x^{n} + a_{n-1}x^{n-1}+ \ldots + a_1x + a_0,

    where all a_j are even and a_0 = 2^m + (k+1)(a-1) \not \equiv 0 \mod 4.
    There is condition in the text of the question a_k=a_{k-1}=\ldots =a_1=a_0=a , where a is an odd number so a_j cannot be even number.
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    Re: Irreducible polynomials over ring of integers ?

    Quote Originally Posted by princeps View Post
    There is condition in the text of the question a_k=a_{k-1}=\ldots =a_1=a_0=a , where a is an odd number so a_j cannot be even number.
    in my solution, a_j are the coefficients of f(x+1) not f(x).
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  5. #5
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    Re: Irreducible polynomials over ring of integers ?

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