Degree of divisors on curves

Question

When Hartshorne talked about divisors on (noetherian, integral, separated, regular in codimension 1) curves (this is on page 137), he defined the degree of $\sum n_iP_i$ (where $P_i$ are closed points) to be $\sum n_i$, but shouldn't degree of a divisor be $\sum n_i \textrm{ deg }P_i$? And when the field is not algebraically closed, the closed points may not have degree 1?

Answer 1

In fact, the definition for the degree of a divisor on a projective variety is in page 132. When D is the divisor of a curve then we can say like you, in page 137.