Imaginary Hyperelliptic Curve - Coordinate Ring

Coordinate Ring

The coordinate ring of C over K is defined as

.

The polynomial is irreducible over, so

is an integral domain.
Proof. If r (x,y) were reducible over, it would factor as (y - u(x)) · (y - v(x)) for some u,v ∈ . But then u(x) · v(x)= f(x) so it has degree 2g + 1, and u(x) + v(x) = h(x) so it has degree smaller than g, which is impossible.

Note that any polynomial function can be written uniquely as

with, ∈

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