In this section, we introduce some elementary concepts from algebraic geometry, which facilitate the treatment of elliptic and hyperelliptic curves in the next two sections. We concentrate only on plane curves, because these are the only curves we need in this book. Throughout this section, K denotes a field (finite or infinite) and the algebraic closure of K.

The solutions of a polynomial equation f(X, Y) = 0 is one of the central objects of study in algebraic geometry. For example, we know that in the equation X^{2} + Y^{2} – 1 = 0 represents a circle with origin at (0, 0) and with radius ...

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