4
$\begingroup$

I'm looking for the smallest positive integer $n$ such that there's a quartic polynomial in $\Bbb Z_n [X]$ that has 8 distinct roots in $\Bbb Z_n$.

I have n equal to 15 with roots 1, 2, 4, 7, 8, 11, 13, 14 and equation $x^4-1=0$ but I'm unsure if you can go lower (is it possible to have n equal to 8?)

Also, while my math experience barely touches the surface of number theory and abstract algebra thus I'm not necessarily looking to prove this, I would like to understand why my claim of the minimality of n is true (or why yours for n less than 15 is) which I currently do not.

$\endgroup$
  • $\begingroup$ $8$ distinct roots? $\endgroup$ – Bernard Oct 7 '15 at 23:44
  • 1
    $\begingroup$ Yes; 8 distinct roots $\endgroup$ – Liam Cooney Oct 7 '15 at 23:44
6
$\begingroup$

Look at $4x(x+1)$, in $\mathbb{Z}_8$. Oops, a quadratic! Multiply by something.

For another example, this time monic, use $x(x+1)(x+2)(x+3)$, again in $\mathbb{Z}_8$.

$\endgroup$
  • $\begingroup$ Yeah, I was thinking in monic polynomials. $\endgroup$ – ajotatxe Oct 7 '15 at 23:52
  • $\begingroup$ I am literally minded! $\endgroup$ – André Nicolas Oct 7 '15 at 23:53
  • $\begingroup$ Golly gee I wonder what (this is as easy as I think right...?). Thank you! Damn easy to justify too ;) $\endgroup$ – Liam Cooney Oct 7 '15 at 23:58
  • $\begingroup$ Yes, pretty easy, of any two consecutive evens, one is divisible by $4$. $\endgroup$ – André Nicolas Oct 7 '15 at 23:59
  • $\begingroup$ I meant you can easily just multiply your cubic by x to get a quint of function that works $\endgroup$ – Liam Cooney Oct 8 '15 at 0:01
2
$\begingroup$

For monic examples, try $f(x) = x^4 + 2 x^3 + 7 x^2 + 6 x$ or $x^4 + 6 x^3 + 3 x^2 + 6 x$ in $\mathbb Z_8$.

EDIT: And at the next level, there are monic sextics over $\mathbb Z_{16}$ with $16$ different roots. For example, ${x}^{6}+3\,{x}^{5}+9\,{x}^{4}+5\,{x}^{3}+6\,{x}^{2}+8\,x$.

$\endgroup$
  • 1
    $\begingroup$ How are all you brilliant human beings coming up with these? Is there any real significance to it being monic or not monic? I'm just now learning what monic means... $\endgroup$ – Liam Cooney Oct 8 '15 at 0:00
  • $\begingroup$ Obviously it's easier to do in $\mathbb Z_8$ if all the coefficients are even. Multiplying everything by an odd number, on the other hand, doesn't change anything. $\endgroup$ – Robert Israel Oct 8 '15 at 0:39
1
$\begingroup$

Every number raied to $4$ ends in the same digit than the number itself, so $X^4-X$ has 10 roots in $\Bbb Z_{10}$, so $15$ is not the answer.

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.