References
-
P. Barrucand and H. Cohn, Note on primes of type
$x^{2}+32y^{2}$ , class number, and residuacity, J. Reine Angew. Math. 238 (1969), 67-70. - P. E. Conner and J. Hurrelbrink, Class Number Parity, Ser. Pure Math. 8, Would Sci., Singapore 1988.
-
P. E. Conner and J. Hurrelbrink, On the 4-rank of the tame kernel
$K_{2}$ (O) in positive definite terms, J. Number Theory 88 (2001), no. 2, 263-282. https://doi.org/10.1006/jnth.2000.2626 - F. Gerth III, Counting certain number fields with prescibed l-class numbers, J. Reine Angew. Math. 337 (1982), 195-207.
- F. Gerth III, The 4-class ranks of quadratic fields, Invent. Math. 77 (1984), no. 3, 489-515. https://doi.org/10.1007/BF01388835
- F. Gerth III and S. W. Graham, Application of a character sum estimate to a 2-class number density, J. Number Theory 19 (1984), no. 2, 239-247. https://doi.org/10.1016/0022-314X(84)90108-2
- G. Hardy and E. Wright, An Introduction to the Theory of Numbers, Fifth edition, London, 1979.
- E. Hecke, Lecture on the Theory of Algebraic Numbers, GTM 77, Springer-Verlag, 1981.
-
J. Hurrelbrink and Q. Yue, On ideal class groups and units in terms of the quadratic form
$x^{2}+32y^{2}$ , Chinese Ann. Math. Ser. B 26 (2005), no. 2, 239-252. https://doi.org/10.1142/S0252959905000208 - K. Ireland and M. Rosen, A Classical Introduction to Modern Number Theory, GTM 84, Springer-Verlag, 1972.
- J. Neukirch, Class Field Theory, Springer, Berlin, 1986.
- P. Stevenhagen, Divisibity by 2-powers of certain quadratic class numbers, J. Number Theory 43 (1993), no. 1, 1-19. https://doi.org/10.1006/jnth.1993.1001
- X. Wu and Q. Yue, 8-ranks of class groups of some imaginary quadratic number fields, Acta Math. Sin. (Engl. Ser.) 23 (2007), no. 11, 2061-2068. https://doi.org/10.1007/s10114-007-0965-1
- Q. Yue, On tame kernel and class group in terms of quadratic forms, J. Number Theory 96 (2002), no. 2, 373-387. https://doi.org/10.1016/S0022-314X(02)92808-8
- Q. Yue, 8-ranks of class groups of quadratic number fields and their densities, Acta Matematica Sinica (Eng. Ser.), to apppear.
- Q. Yue and J. Yu, The densities of 4-ranks of tame kernels for quadratic fields, J. Reine Angew. Math. 567 (2004), 151-173.
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