Comparison of 4-class ranks of certain quadratic fields

Gerth, Frank, III

doi:10.1090/S0002-9939-01-05922-6

Comparison of 4-class ranks of certain quadratic fields

Author: Frank Gerth III
Journal: Proc. Amer. Math. Soc. 129 (2001), 2547-2552
MSC (2000): Primary 11R11, 11R29, 11R45
DOI: https://doi.org/10.1090/S0002-9939-01-05922-6
Published electronically: January 23, 2001
MathSciNet review: 1838376
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Abstract: Let be a square-free positive integer. Let $r_{4}(K)$ denote the 4-class rank of a quadratic field . This paper examines how likely it is for $r_{4}(\mathbb{Q} (\sqrt {-m}\,)) =r_{4} (\mathbb{Q} (\sqrt {m}\,))$ and for $r_{4} (\mathbb{Q} (\sqrt {-m}\,)) = r_{4} (\mathbb{Q} (\sqrt {m}\,)) +1$ .

1. J. Cremona and R. Odoni, Some density results for negative Pell equations; an application of graph theory, J. London Math. Soc. 39 (1989), 16-28. MR 90b:11019
2. P. Damey and J. Payan, Existence et construction des extensions galoisiennes et non-abéliennes de degré 8 d'un corps de caractéristique différente de 2, J. Reine Angew. Math. 244 (1970), 37-54. MR 43:6186
3. F. Gerth, Counting certain number fields with prescribed $\ell$ -class numbers, J. Reine Angew. Math. 337 (1982), 195-207. MR 84c:12002
4. F. Gerth, The 4-class ranks of quadratic fields, Invent. Math. 77 (1984), 489-515. MR 85j:11137
5. F. Gerth, Densities for ranks of certain parts of -class groups, Proc. Amer. Math. Soc. 99 (1987), 1-8. MR 88b:11067
6. F. Halter-Koch, Über den 4-Rank der Klassengruppe quadratischer Zahlkörper, J. Number Theory 19 (1984), 219-227. MR 86a:11041