Article
| Title: | Ideal class groups of some quadratic number fields and factorization of values of some quadratic polynomials (English) |
| Author: | Louboutin, Stéphane R. |
| Language: | English |
| Journal: | Czechoslovak Mathematical Journal |
| ISSN: | 0011-4642 (print) |
| ISSN: | 1572-9141 (online) |
| Volume: | 76 |
| Issue: | 2 |
| Year: | 2026 |
| Pages: | 677-697 |
| Summary lang: | English |
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| Category: | math |
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| Summary: | We fill the gaps in Gica's determination of all the odd positive integers $d$ for which the number of distinct prime divisors of $f_d(x)=d+x^2$ is less than or equal to $2$ for all positive and odd integers $x\leq \sqrt {d}$. We also determine all the even positive integers $d$ for which the number of distinct prime divisors of $f_d(x)$ is less than or equal to $2$ for all positive and even integers $x\leq \sqrt {d}$. These problems are related to famous Frobenius-Rabinowitsch's characterization of the imaginary quadratic number fields ${\mathbb Q}(\sqrt {-d})$ of odd discriminants with class number one in terms of the primality of $\frac 14 f_d(x)$ for all positive and odd integers $x\leq \sqrt {d}$. However, the solution to our problem is much more difficult to come up with. We also begin to address the same problems for the case of $f_d(x)=d-x^2$, in relation with the class groups of real quadratic number fields ${\mathbb Q}(\sqrt {d})$. (English) |
| Keyword: | quadratic field |
| Keyword: | imaginary quadratic field |
| Keyword: | class group |
| Keyword: | class number |
| Keyword: | quadratic polynomial |
| Keyword: | Frobenius-Rabinowitsch |
| MSC: | 11R11 |
| MSC: | 11R27 |
| MSC: | 11R29 |
| DOI: | 10.21136/CMJ.2026.0455-25 |
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| Date available: | 2026-05-22T11:25:15Z |
| Last updated: | 2026-05-25 |
| Stable URL: | http://hdl.handle.net/10338.dmlcz/153656 |
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