I read the definition of the group law on elliptic curves and there's one thing I don't understand. In the link above it is stated that
In the projective plane, each line will intersect a cubic at three points when accounting for multiplicity.
In an algebraically closed field this is a consenquence of Bézout's theorem, but is it true if the elliptic curve is defined over a field which is not necessarily algebraically closed? If not, how is the group law defined over such fields?