An elementary proof of the theorem on the imaginary quadratic fields with class number 1
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2024
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| _version_ | 1866917567436685312 |
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| author | Carter, James E. |
| author_facet | Carter, James E. |
| contents | Let $D$ be a square-free integer other than 1. Let $K$ be the quadratic field ${\mathbb Q}(\sqrt D)$. Let $δ\in \{1,2\}$ with $δ=2$ if $D\equiv 1 \pmod 4$. To each prime ideal $\mathcal P$ in $K$ that splits in $K/\mathbb Q$ we associate a binary quadratic form $f_{\mathcal P}$ and show that when $K$ is imaginary then $\mathcal P$ is principal if and only if $f_{\mathcal P}$ represents $δ^2$, and when $K$ is real then $\mathcal P$ is principal if and only if $f_{\mathcal P}$ represents $\pm δ^2$. As an application of this result we obtain an elementary proof of the well-known theorem on the imaginary quadratic fields with class number 1. The proof reveals some new information regarding necessary conditions for an imaginary quadratic field to have class number 1 when $D\equiv 1 \pmod 4$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_07192 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | An elementary proof of the theorem on the imaginary quadratic fields with class number 1 Carter, James E. Number Theory 11R29 (Primary) 11R04, 11R11, 11D09 (Secondary) Let $D$ be a square-free integer other than 1. Let $K$ be the quadratic field ${\mathbb Q}(\sqrt D)$. Let $δ\in \{1,2\}$ with $δ=2$ if $D\equiv 1 \pmod 4$. To each prime ideal $\mathcal P$ in $K$ that splits in $K/\mathbb Q$ we associate a binary quadratic form $f_{\mathcal P}$ and show that when $K$ is imaginary then $\mathcal P$ is principal if and only if $f_{\mathcal P}$ represents $δ^2$, and when $K$ is real then $\mathcal P$ is principal if and only if $f_{\mathcal P}$ represents $\pm δ^2$. As an application of this result we obtain an elementary proof of the well-known theorem on the imaginary quadratic fields with class number 1. The proof reveals some new information regarding necessary conditions for an imaginary quadratic field to have class number 1 when $D\equiv 1 \pmod 4$. |
| title | An elementary proof of the theorem on the imaginary quadratic fields with class number 1 |
| topic | Number Theory 11R29 (Primary) 11R04, 11R11, 11D09 (Secondary) |
| url | https://arxiv.org/abs/2401.07192 |