The $n$-th cyclotomic polynomial $\Phi_{n,K}(x)$ associated with a field $K$ (with characteristic 0 or coprime with $n$) is $\prod_{1\leq j\leq n, (j,n) = 1}(x - \omega^j)$ where $\omega$ is a generator of the group $\mu_n(L) = \{\alpha \in L: \alpha^n = 1\}$, with $L$ being a splitting field of $x^n - 1 \in K[x]$. Let's note the well-known fact that $\Phi_{n,K} \in K[x]$.
My question is - does the polynomial essentially depend on the field? What I mean is - for example $\Phi_{8,\mathbb{Q}} = x^4 + 1$ but it's looks to me that (for $p$ - odd) $\Phi_{8,\mathbb{F}_p} = x^4 + 1$? There is only the formal difference of the coefficients belonging to $\mathbb{Q}$ in one case and to $\mathbb{F}_p$ in the other. But the polynomials "look the same" - more formally: $\Phi_{8,\mathbb{Q}} \pmod p = \Phi_{8,\mathbb{F}_p}$. Is this true for all $n$ and why?