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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.

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

I suppose no, here are some examples:

$$ x^{3} + x^{2} + x + 1 = (x + 1) \cdot (x^{2} + 1) $$ $$ x^{3} + x^{2} + x + 6 = (x + 2) \cdot (x^{2} - x + 3) $$ $$ x^{3} + x^{2} + x + 21 = (x + 3) \cdot (x^{2} - 2x + 7) $$ $$ x^{3} + x^{2} + x + 52 = (x + 4) \cdot (x^{2} - 3x + 13) $$ $$ x^{3} + x^{2} + x + 1 = (x + 1) \cdot (x^{2} + 1) $$ $$ x^{4} + x^{3} + x^{2} + x + 12 = (x^{2} - 2x + 3) \cdot (x^{2} + 3x + 4) $$ $$ x^{4} + x^{3} + x^{2} + 12x + 12 = (x + 2) \cdot (x^{3} - x^{2} + 3x + 6) $$ $$ x^{5} + x^{4} + x^{3} + x^{2} + x + 1 = (x + 1) \cdot (x^{2} - x + 1) \cdot (x^{2} + x + 1) $$ $$ x^{5} + x^{4} + x^{3} + x^{2} + x + 22 = (x + 2) \cdot (x^{4} - x^{3} + 3x^{2} - 5x + 11) $$

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  • $\begingroup$ you are right.what if $ a_1 \neq 1 $ and $a$ is odd number greater than $1$ ? $\endgroup$ – pedja Nov 10 '11 at 9:20
  • $\begingroup$ You mean $k>1$ and $a>1$ odd ? There is an example with $a=21$ $\endgroup$ – joro Nov 10 '11 at 9:24
  • $\begingroup$ no, $a_1$ is coefficient of the linear term...example: $ x^2+3x+3 $ where $ a_1=a=3 $ $\endgroup$ – pedja Nov 10 '11 at 9:29
  • $\begingroup$ in example above $a_0=a=21$ $\endgroup$ – pedja Nov 10 '11 at 9:32
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    $\begingroup$ 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 $a_{73}\gt1089$? $\endgroup$ – Gerry Myerson Nov 10 '11 at 11:27

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