In Milne's algebraic number theory notes, on page 65, there is the following example:
$X^4+X^3+X^2+X+1\equiv(X+4)^4\pmod{5}$.
And Milne asks: Why is that obvious?
This comes after discussion of ramification of prime ideals in number fields.
My attempt so far:
Let $\omega_5=e^{2\pi i/5}$. The field extension $\mathbb{Q}\subset\mathbb{Q}(\omega_5)=K$ is Galois and of degree $4$. The minimal polynomial of $\omega_5$ over $\mathbb{Q}$ is $f(X)=X^4+X^3+X^2+X+1$.
Consider prime ideal $(5)\lhd\mathbb{Z}$ and its factorization in the ring of integers $\mathcal{O}_K$.
I'm not sure how to proceed.