What are some simple proofs that the points on an elliptic curve form an abelian group under addition? I am mostly looking for proofs of closure and associativity, since the other three requirements follow immediately from the definitions. Note that I am specifically looking for simpler proofs, since I am only still in high-school, so I have not learned about homomorphisms and isomorphisms and matrices yet. An intuitive 'proof' will also suffice (since again, it's only high-school level). Also, are elliptic curve points always an abelian group under addition? Or only over the rational numbers or over some finite prime field (and not over the reals)? I'm trying to write a paper/essay on elliptic curve cryptography, but I can't get these proofs to work.
2026-03-29 02:36:16.1774751776
Proof of elliptic curves being an abelian group
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