# Are these polynomials irreducible over ring Z of integers ?

Is it true that polynomials of the form :

$f(x)= x^n+x^{n-1}+...+x^{k+1}+ax^k+ax^{k-1}+...a$

where $gcd(n+1,k+1)=1$ and $a\in \mathbb{Z^{+}}$

are irreducible over ring $\mathbb{Z}$ of integers ?

Neither of Eisenstein's criterion and Cohn's criterion cannot be applied on the polynomials of this form. I have checked a lot of cases and it seems to be true.

• Did you mean to exclude a=1? If not, this will not be irreducible whenever n+1 is composite. – Harry Altman Nov 10 '11 at 9:04
• Counterexample: $x^2 + 4x + 4$. – S. Carnahan Nov 10 '11 at 9:08
• @Carnahan,what if a is odd and greater than 1 – pedja Nov 10 '11 at 9:10
• The third example is with $a=21$ pedja. – joro Nov 10 '11 at 9:25

Probably you want $a \ne 1$.
$$x^{3} + x^{2} + x + 1 = (x + 1) \cdot (x^{2} + 1)$$ $$x^{3} + x^{2} + x + 6 = (x + 2) \cdot (x^{2} - x + 3)$$ $$x^{3} + x^{2} + x + 21 = (x + 3) \cdot (x^{2} - 2x + 7)$$ $$x^{3} + x^{2} + x + 52 = (x + 4) \cdot (x^{2} - 3x + 13)$$ $$x^{3} + x^{2} + x + 1 = (x + 1) \cdot (x^{2} + 1)$$ $$x^{4} + x^{3} + x^{2} + x + 12 = (x^{2} - 2x + 3) \cdot (x^{2} + 3x + 4)$$ $$x^{4} + x^{3} + x^{2} + 12x + 12 = (x + 2) \cdot (x^{3} - x^{2} + 3x + 6)$$ $$x^{5} + x^{4} + x^{3} + x^{2} + x + 1 = (x + 1) \cdot (x^{2} - x + 1) \cdot (x^{2} + x + 1)$$ $$x^{5} + x^{4} + x^{3} + x^{2} + x + 22 = (x + 2) \cdot (x^{4} - x^{3} + 3x^{2} - 5x + 11)$$
• you are right.what if $a_1 \neq 1$ and $a$ is odd number greater than $1$ ? – pedja Nov 10 '11 at 9:20
• You mean $k>1$ and $a>1$ odd ? There is an example with $a=21$ – joro Nov 10 '11 at 9:24
• no, $a_1$ is coefficient of the linear term...example: $x^2+3x+3$ where $a_1=a=3$ – pedja Nov 10 '11 at 9:29
• in example above $a_0=a=21$ – pedja Nov 10 '11 at 9:32
• pedja, I think at this point the onus is on you to either find an example yourself or to give some reason why you think there may not be one. Otherwise, I can see an infinity of questions...what if $a_2\gt1$? what if $a_3\gt1$?... what if $a_{73}\gt1089$? – Gerry Myerson Nov 10 '11 at 11:27