I have some troubles solving the following problem:
Let $E$ be the elliptic curve $E:y^2+2y=x^3+x+9$ over $\mathbb{F}_{16}$. Compute the 2-torsion group $E[2]$, i.e. find all the points of order $2$ on $E$.
My idea is: if $2P=0$ then $P=-P$, so I have to find all the points that have this property. Usually if $P=(x,y)$ then $-P$ is defined as $-P=(x,-y-a_{1}x-a_{3})$. In our case $-P$ becomes $-P=(x,-y-2)$.
Can I say that since the characteristic of $\mathbb{F}_{16}$ is $2$ then $2y=0$ and $-y=y$ and then my equation becomes $E:y^2=x^3+x+9$ and $-P$ becomes $-P=(x,y)$? If this is correct, i'm right if I say that then obviously $P=-P$ for each point on the curve and so the 2-torsion group is simply $E(\mathbb{F}_{16})$?