Proposition:

The ideal class group of hyperelliptic function fields is finite.

Proof:

Let F be a hyperelliptic function field of the form F = K(x,y), where K is a field and y^2 = f(x) is a polynomial of degree d > 2 with coefficients in K. Let Cl(F) be the ideal class group of F.

We know that the genus of F is (d-1)/2, and F has d+1 places at infinity.

We will first show that the divisor class group of F, denoted by Pic(F), is finite. The Picard group of F is isomorphic to the Jacobi group of the curve defined by y^2 = f(x). The Jacobi group is a finite abelian group, and therefore Pic(F) is also a finite abelian group.

Let D be a divisor of F. We can write D as D = div(f) + D', where div(f) is the divisor of the function f, and D' is a divisor of F not containing any pole of f. Since f has only finitely many zeros and poles, it follows that there are only finitely many divisors of the form div(f) + D', and hence Pic(F) is a finite group.

Now, we will show that Cl(F) is a quotient group of Pic(F) and hence is also finite. Let I be an ideal of F. We define the principal divisor associated to I as (I) = {f in F | v_P(f) >= v_P(I) for all places P of F}. We note that (I) is a divisor of F.

Two ideals I and J are equivalent if and only if their associated principal divisors (I) and (J) are linearly equivalent, i.e., (I) - (J) is a principal divisor. Therefore, the ideal class group Cl(F) is isomorphic to the quotient group Pic(F)/Prin(F), where Prin(F) is the subgroup of Pic(F) consisting of principal divisors.

Since Pic(F) is finite, and Prin(F) is a finite subgroup of Pic(F), it follows that Cl(F) is also a finite group. Hence, the ideal class group of hyperelliptic function fields is finite.

Therefore, the proposition is true: the ideal class group of hyperelliptic function fields is finite.