| Title:
|
Lower bound for class numbers of certain real quadratic fields (English) |
| Author:
|
Mishra, Mohit |
| Language:
|
English |
| Journal:
|
Czechoslovak Mathematical Journal |
| ISSN:
|
0011-4642 (print) |
| ISSN:
|
1572-9141 (online) |
| Volume:
|
73 |
| Issue:
|
1 |
| Year:
|
2023 |
| Pages:
|
1-14 |
| Summary lang:
|
English |
| . |
| Category:
|
math |
| . |
| Summary:
|
Let $d$ be a square-free positive integer and $h(d)$ be the class number of the real quadratic field $\mathbb {Q}{(\sqrt {d})}.$ We give an explicit lower bound for $h(n^2+r)$, where $r=1,4$. Ankeny and Chowla proved that if $g>1$ is a natural number and $d=n^{2g}+1$ is a square-free integer, then $g \mid h(d)$ whenever $n>4$. Applying our lower bounds, we show that there does not exist any natural number $n>1$ such that $h(n^{2g}+1)=g$. We also obtain a similar result for the family $\mathbb {Q}(\sqrt {n^{2g}+4})$. As another application, we deduce some criteria for a class group of prime power order to be cyclic. (English) |
| Keyword:
|
real quadratic field |
| Keyword:
|
class group |
| Keyword:
|
class number |
| Keyword:
|
Dedekind zeta values |
| MSC:
|
11R11 |
| MSC:
|
11R29 |
| MSC:
|
11R42 |
| idZBL:
|
Zbl 07655753 |
| idMR:
|
MR4541087 |
| DOI:
|
10.21136/CMJ.2022.0264-21 |
| . |
| Date available:
|
2023-02-03T11:06:02Z |
| Last updated:
|
2025-04-07 |
| Stable URL:
|
http://hdl.handle.net/10338.dmlcz/151500 |
| . |
| Reference:
|
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