I am studying elliptic curves. From the very underlying group to groups of n-torsion points, free modules are all around.
For example you read Elliptic Curves: Number Theory and Cryptography (By Lawrence C. Washington) and AEC (by J.H. Silverman) you find they sometimes feel free to complete a basis given a first point. I know in a vector space this can always be done (you can do division in a field). But I really don't understand why you could do this for, say, n-torsion of a EC on a finite field F (n doesn't divide char(F)).
What Am I missing? It seems to me that being linear indipendent for two points is not sufficient to span all the group.
EXAMPLE: page 400 AEC 2nd edition, proof of Theorem 9.4 "Choose another N-torsion point T' [...]". How?
EXAMPLE 2: page 158 Washington, Proof of Prosition 5.5. Here he can find a basis because N is prime so E[N] is a vector field.
A non-zero element of $(\mathbb{Z}/N)^2$ is not necessarily even linearly independent by itself but if it does then it can always be completed to a basis, see eg https://math.stackexchange.com/a/4634212/32766.