11. Quadratic fields have been studied in great depth, initially as part of the theory of binary quadratic forms.12. Already Gauss had shown that, in fact, every quadratic field is contained in a larger cyclotomic field. 13. Dually, an imaginary quadratic field ) } } admits no real embeddings and a conjugate pair of complex embeddings. 14. Those L-functions are for the Dirichlet characters which are the Jacobi symbols attached to the three quadratic fields . 15. *Let " K " be an imaginary quadratic field whose discriminant has at least 6 prime factors. 16. A real quadratic field ) } } is so-called because it admits two real embeddings and no complex embeddings. 17. The Cohen-Lenstra heuristics are a set of more precise conjectures about the structure of class groups of quadratic fields . 18. There is exactly one quadratic field for every fundamental discriminant " D " 0 ` " 1, up to isomorphism. 19. This description was used by Sarnak to establish a conjecture of Gauss on the mean order of class groups of real quadratic fields . 20. Thus, for example, the only fields for which the rank of the free part is zero are and the imaginary quadratic fields .
