I am trying to understand what an elliptic curve mod $\pi$ vs mod $p$ is. Basically I am confused about the treatment given in Silverman's two books.
The definition for mod $p$ in Rational Points of Elliptic Curves is basically sending the group of points $C(\mathbb{Q})$ to $C(\mathbb{F}_p)$. From my understanding, $C$ has a good reduction if $C/\mathbb{F}_p$ is not singular, i.e. $p$ does not divide the discriminant of the new equation.
But in Arithmetic of Elliptic Curves they introduce a lot of other propositions such as "Let $E/K$ be an elliptic curve. Then E has potential good reduction if and only if its $j$-invariant is integral, i.e., if and only if $j(E) ∈ R$". ($R$ being the ring of integers of $K$).
I was really confused because I can't understand what $R$ would be for the field $\mathbb{F}_p$ until I read the definition of algebraic number field. So I don't think $\mathbb{F}_p$ is a number field...so my question is what is going on with these two definitions? Are they just different levels of complexity or is it just something completely different about local fields?