Why coordinates of elliptic curves's points are integers?

If you look at elliptic curves over CC, and do point addition with points with integer coordinates (as Z⊂CZ⊂C), then the result of the point addition usually will not have integer coordinates.

But in cryptography, we don't use elliptic curves over CC, but over a finite field FF. So the coordinates are not integers, but field elements. Subtracting and dividing field elements by each other (assuming no division by zero) gives you new field elements.

If you represent field elements by integers, then the new coordinates are "integers" again.

The question would be asked as why the coordinates of the points of an elliptic curve definite over a finit field are elements of that finit field? .