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Given a field (K,+,·)
[usually denoted just by K],
an elliptic curve over K [denoted by E(K)] is the set of ... - {(x,y) ∈ K2 | p(x,y)=0},
where "p" is a 3-degree polynomial with coefficients in K, and it has no singular point
(cusps or self-intersections)
∪
- &inf; (called "infinity") ∉ K2
K may be an infinite (ℚ,
ℝ,
&complex;, ...)
or a finite (&cuerpoFinito;q)
[as &modulos;p,
with p prime]
set.
Note that: K can't be ℕ,
ℤ,
or &modulos;p [p not prime]
because they are not fields (they don't have multiplicative inverse).
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