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