Smallest n such that there's a polynomial in $\Bbb Z_n [X]$ of degree 4 that has 8 roots in $\Bbb Z_n$

Question

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.

Answer 1

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 $\mathbb{Z}_8$.

Answer 2

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.

Answer 3

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, ${x}^{6}+3\,{x}^{5}+9\,{x}^{4}+5\,{x}^{3}+6\,{x}^{2}+8\,x$.