= Solution
Let $O=(0:1:0)$ be the point at infinity of the <Weierstrass equation of an elliptic curve>. It is the <identity element> for the <elliptic curve group law>. For an affine <rational point> $P=(x,y)$, the inverse is
$$
-P=(x,-y-a_1x-a_3).
$$
Indeed, the two affine intersections with the vertical line have these ordinates, since their sum as roots of the quadratic in $y$ is $-a_1x-a_3$. This reflection is valid for a generalized <Weierstrass equation of an elliptic curve> in every <characteristic of a field>.
For distinct $P,Q$, draw their line; for $P=Q$, draw the <tangent line>. Let $R$ be its third intersection with the <elliptic curve>, counted with <intersection multiplicity>. Then \b[the sum is the reflection of the third intersection]:
$$
\boxed{P+Q=-R.}
$$
A vertical line has third intersection $O$, so $P+(-P)=O$. If a tangent has third intersection $O$, then $2P=O$. Define addition with $O$ by $P+O=P$; these conventions include all exceptional cases of the affine construction.
The construction is defined over the coefficient <field>: after two known intersections, the remaining root is rational over that <field>. It gives a commutative <group operation>. To see why it is associative, use the <elliptic curve group law from Riemann-Roch>: the map $P\mapsto[P-O]$ identifies the <elliptic curve> with its degree-zero <divisor class group>. A line has intersection divisor $P+Q+R$ <linearly equivalent> to $3O$, the divisor of the line at infinity. Thus $[P-O]+[Q-O]+[R-O]=0$. The geometric sum agrees with addition of <divisor classes>, which is associative.
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