On the Class Numbers of Algebraic Number Fields
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| 024 | 7 | 0 | |a 10.1007/s10958-015-2416-3 |2 doi |
| 035 | |a (NATIONALLICENCE)springer-10.1007/s10958-015-2416-3 | ||
| 100 | 1 | |a Fomenko |D O. |u St. Petersburg Department of the Steklov Mathematical Institute, St. Petersburg, Russia |4 aut | |
| 245 | 1 | 0 | |a On the Class Numbers of Algebraic Number Fields |h [Elektronische Daten] |c [O. Fomenko] |
| 520 | 3 | |a Let K be a number field of degree n over ℚ and let d, h, and R be the absolute values of the discriminant, class number, and regulator of K, respectively. It is known that if K contains no quadratic subfield, then h R ≫ d 1 / 2 log d , $$ h\;R\gg \frac{d^{1/2}}{ \log d}, $$ where the implied constant depends only on n. In Theorem 1, this lower estimate is improved for pure cubic fields. Consider the family K n $$ {\mathcal{K}}_n $$ , where K ∈ K n $$ {\mathcal{K}}_n $$ if K is a totally real number field of degree n whose normal closure has the symmetric group S n as its Galois group. In Theorem 2, it is proved that for a fixed n ≥ 2, there are infinitely many K ∈ K n $$ {\mathcal{K}}_n $$ with h ≫ d 1 / 2 log log d n − 1 / log d n , $$ h\gg {d}^{1/2}{\left( \log \log d\right)}^{n-1}/{\left( \log d\right)}^n, $$ where the implied constant depends only on n. This somewhat improves the analogous result h ≫ d 1/2/(log d) n of W. Duke [MR 1966783 (2004g:11103)]. | |
| 540 | |a Springer Science+Business Media New York, 2015 | ||
| 773 | 0 | |t Journal of Mathematical Sciences |d Springer US; http://www.springer-ny.com |g 207/6(2015-06-01), 934-939 |x 1072-3374 |q 207:6<934 |1 2015 |2 207 |o 10958 | |
| 856 | 4 | 0 | |u https://doi.org/10.1007/s10958-015-2416-3 |q text/html |z Onlinezugriff via DOI |
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| 900 | 7 | |a Metadata rights reserved |b Springer special CC-BY-NC licence |2 nationallicence | |
| 908 | |D 1 |a research-article |2 jats | ||
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| 950 | |B NATIONALLICENCE |P 856 |E 40 |u https://doi.org/10.1007/s10958-015-2416-3 |q text/html |z Onlinezugriff via DOI | ||
| 950 | |B NATIONALLICENCE |P 100 |E 1- |a Fomenko |D O. |u St. Petersburg Department of the Steklov Mathematical Institute, St. Petersburg, Russia |4 aut | ||
| 950 | |B NATIONALLICENCE |P 773 |E 0- |t Journal of Mathematical Sciences |d Springer US; http://www.springer-ny.com |g 207/6(2015-06-01), 934-939 |x 1072-3374 |q 207:6<934 |1 2015 |2 207 |o 10958 | ||