Thread:

1. 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.

2. Re: Irreducible polynomials over ring of integers ?

Originally Posted by princeps
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.

3. Re: Irreducible polynomials over ring of integers ?

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

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.

4. Re: Irreducible polynomials over ring of integers ?

Originally Posted by princeps
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 $f(x)$.

5. Re: Irreducible polynomials over ring of integers ?

Can you give me an example of f(x+1) ?
{"extraction_info": {"found_math": true, "script_math_tex": 0, "script_math_asciimath": 0, "math_annotations": 0, "math_alttext": 0, "mathml": 0, "mathjax_tag": 0, "mathjax_inline_tex": 0, "mathjax_display_tex": 0, "mathjax_asciimath": 0, "img_math": 0, "codecogs_latex": 50, "wp_latex": 0, "mimetex.cgi": 0, "/images/math/codecogs": 0, "mathtex.cgi": 0}, "config": {"markdown_headings": false, "markdown_code": true, "boilerplate_config": {"ratio_threshold": 0.18, "absolute_threshold": 10, "end_threshold": 15, "enable": true}, "remove_buttons": true, "remove_image_figures": true, "remove_link_clusters": true, "table_config": {"min_rows": 2, "min_cols": 3, "format": "github"}, "remove_chinese": true, "remove_edit_buttons": true, "extract_latex": true}, "english_confidence": 0.8640118837356567, "perplexity_flag": "head"}