-1
$\begingroup$

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

$\endgroup$

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

  • This question does not appear to be about cryptography within the scope defined in the help center.
If this question can be reworded to fit the rules in the help center, please edit the question.

  • 1
    $\begingroup$ I'm voting to close this question as off-topic because it is about general mathematics. $\endgroup$ – fkraiem Apr 5 '17 at 7:21
3
$\begingroup$

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.

$\endgroup$

Not the answer you're looking for? Browse other questions tagged or ask your own question.