กค On class numbers of quadratic extensions
Class numbers are already deep for quadratic extensions of Q. There is a question of Gauss, which may be phrased as:
[Problem] For n>=1, describe the set of all quadratic extensions F such that the class number h_F = n.
There is a sharp contrast between the real quadratic and imaginary quadratic fields. In the latter case, we have
[Theorem] (Heilbronn, 1930s) As the discriminant tends to negative infinity, h_F tends to positive infinity.
[Theorem] (Heegner, Stark, 1960s) For n=1, the complete list of all discriminants for such imaginary quadratic fields is
[Theorem] (Goldfeld, 1980s) For any n, the set of all imaginary F with h_F=n can be effectively determined. (For instance, the algorithm can be implemented into a computer program.)
However, little is known in the real quadratic case. In fact, it is not believed that h_F tends to infinity as the disc -> positive infinity. You may become famous if you can settle
[Conjecture] (Gauss) There are infinitely many real quadratic fields whose class numbers are one.