These modular forms, for real quadratic fields, were first treated in the 1901 G�ttingen University " Habilitationssschrift " of Otto Blumenthal.
22.
The rank is > 0 for all number fields besides "'Q "'and imaginary quadratic fields, which have rank 0.
23.
Binary quadratic forms have been extensively studied in number theory, in particular, in the theory of quadratic fields, continued fractions, and modular forms.
24.
For example, the length of primitive closed geodesics in the modular surface corresponds to the absolute value of units of norm one in real quadratic fields.
25.
Based on his research of the structure of the unit group of quadratic fields, he proved the Dirichlet unit theorem, a fundamental result in algebraic number theory.
26.
*PM : quadratic fields that are not isomorphic, id = 8458 new !-- WP guess : quadratic fields that are not isomorphic-- Status:
27.
*PM : quadratic fields that are not isomorphic, id = 8458 new !-- WP guess : quadratic fields that are not isomorphic-- Status:
28.
According to Galois theory, there must be three quadratic fields contained in " K ", since the Galois group has three subgroups of index 2.
29.
This is the case for which the quadratic field contained in the Galois closure of " K " is the cyclotomic field of cube roots of unity.
30.
Here ? is the quadratic residue symbol modulo " D ", where " " D " is the discriminant of an imaginary quadratic field.
