Algebraic closure for torsion points on elliptic curves In my book about elliptic curves I've read about torsion points of an elliptic curve $E$ defined over $K$, for $n$ positive:
$E[n]=\{P\in E(\overline{K})\mid nP=\infty\}$
I've got two questions about it:


*

*Why do we disregard negative numbers for $n$? The book hasn't defined $K$ to be a residue field or anything of the kind.

*The book emphasizes that $P$ can be in the algebraic closure, but why exactly would they emphasize that? what goes wrong when we don't look at its closure?


Thanks in advance!
 A: 1) We have $E[-n] = E[n]$ for any $n \in \Bbb Z$, so we may assume that $n \geq 0$.
2) Notice that it is possible to look at torsion points over $K$, i.e. $E[n] \cap E(K)$. 
But usually we take points with coordinates in the algebraic closure because we can look at the action of the absolute Galois group of $K$ on $E[n]$ (and its Galois cohomology).
Many properties work better when we look at the algebraic closure. Typically, the multiplication-by-$n$ map is surjective on $\bar K$-points, i.e.
$$[n] : E(\bar K) \to E(\bar K)$$
but it is not surjective on $K$-points, in general (this precisely gives rise to Galois cohomology groups). Compare with the fact that the $n$-th power map $\bar K^{\times} \to \bar K^{\times}$ is surjective, while the $n$-th power map $K^{\times} \to K^{\times}$ is not always surjective.

As for your further comment, the field $\Bbb F_p$ has a unique field extension of degree $n$ for any $n \geq 1$ ; we denote it by $\Bbb F_{p^n}$. Then we may write $\overline{\Bbb F_p}$ as the "union" (or "direct limit") of the fields $\Bbb F_{p^n}$.
