Do every elliptic curve defined over a prime field forms an abelian group?


closed as off-topic by fkraiem, e-sushi Apr 5 '17 at 18:18

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    $\begingroup$ I'm voting to close this question as off-topic because it is about general mathematics. $\endgroup$ – fkraiem Apr 5 '17 at 7:21

Yes, every elliptic curve (that is, smooth projective curve of genus one over a field) admits a natural group structure on its set of points.

(Note: Any nonempty set can be equipped with an abelian group structure, but you are probably only interested in the "nice" group structure on elliptic curves given by rational maps.)

This is a consequence to the Riemann-Roch theorem, which implies that the canonical map $E\to \mathrm{Pic}^0(E)$ is a bijection, and the latter is an abelian group by construction.


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