Can polynomial in Z[x]/(x^n-1) have non-integer coefficients?

Question

I am trying to compute an inverse of some polynomial $f$ in $\mathbb{Z}[x]/(x^5-1)$. Is it possible that $f^{-1}$ has coefficients that are non-integer like 0.33?

Thanks in advance.

Answer 1

Yes, this is possible. For example, $3$ is a polynomial, and the inverse of $3$ is $\frac13$. Or rather, more appropriately, we would say that $3$ has no inverse in $\mathbb{Z}[x] \;/\; (x^5 - 1)$, even though it has an inverse in $\mathbb{Q}[x] \;/\; (x^5 - 1)$.

For another example, $3x$ has the inverse $\frac13 x^4$, so it also doesn't have an inverse with integer coefficients. As a final example, $2x - 1$ has the inverse: $$ \frac{16}{31} x^4 + \frac{8}{31} x^3 + \frac{4}{31} x^2 + \frac{2}{31} x + \frac{1}{31}. $$

All of these are elements with an inverse in $\mathbb{Q}[x] \; / \; (x^5 - 1)$, but not in $\mathbb{Z}[x] \;/\; (x^5 - 1)$.