These investigations of Gauss strongly influenced both the arithmetical theory of quadratic forms in more than two variables and the subsequent development of algebraic number theory, where quadratic fields are replaced with more general number fields.
42.
The Kleinian integers form a ring called the "'Kleinian ring "', which is the ring of integers in the imaginary quadratic field \ mathbb { Q } (-7 ).
43.
Later, C . R . Leedham-Green showed that such an " R " may constructed as the integral closure of a PID in a quadratic field extension ( Leedham-Green 1972 ).
44.
Karl Rubin found a more elementary proof of the Mazur-Wiles theorem by using Kolyvagin's Euler systems, described in and, and later proved other generalizations of the main conjecture for imaginary quadratic fields.
45.
That the zeta function of a quadratic field is a product of the Riemann zeta function and a certain Dirichlet " L "-function is an analytic formulation of the quadratic reciprocity law of Gauss.
46.
Here, PSL denotes the projective special linear group and \ mathcal { O } _ d is the ring of integers of the imaginary quadratic field \ mathbb { Q } ( \ sqrt {-d } ).
47.
A system of elliptic units may be constructed for an elliptic curve " E " with complex multiplication by the ring of integers " R " of an imaginary quadratic field " F ".
48.
Except when the base field is the field of rational numbers or an imaginary quadratic field, the abelian Stark conjectures are still unproved in number fields, and more progress has been made in function fields of an algebraic variety.
49.
With H . Donnelly and I . Singer, he extended Hirzebruch's formula ( relating the signature defect at cusps of Hilbert modular surfaces to values of L-functions ) from real quadratic fields to all totally real fields.
50.
This therefore gives us the precise information about which quadratic field lies in "'Q "'( ? ) . ( That could be derived also by ramification arguments in algebraic number theory; see quadratic field .)
