A note on quadratic cyclotomic extensions
Fuente:
arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866910544975364096 |
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| author | Marques, Sophie Mrema, Elizabeth |
| author_facet | Marques, Sophie Mrema, Elizabeth |
| contents | This paper provides two characterizations of the primitive roots of unity in quadratic cyclotomic extensions over arbitrary fields. Firstly, we introduce a mapping from $\mathbb{N}$ to $\mathbb{N}$ crucial for describing these roots, closely tied to their order over the field. Secondly, for any prime $p$, we determine the maximal natural number $n$ such that $ζ_{p^n}$ defines a quadratic cyclotomic extension over the field $F$. This characterization is uniform across different fields, regardless of their characteristic, and applies to both odd and even primes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2210_03563 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A note on quadratic cyclotomic extensions Marques, Sophie Mrema, Elizabeth Number Theory This paper provides two characterizations of the primitive roots of unity in quadratic cyclotomic extensions over arbitrary fields. Firstly, we introduce a mapping from $\mathbb{N}$ to $\mathbb{N}$ crucial for describing these roots, closely tied to their order over the field. Secondly, for any prime $p$, we determine the maximal natural number $n$ such that $ζ_{p^n}$ defines a quadratic cyclotomic extension over the field $F$. This characterization is uniform across different fields, regardless of their characteristic, and applies to both odd and even primes. |
| title | A note on quadratic cyclotomic extensions |
| topic | Number Theory |
| url | https://arxiv.org/abs/2210.03563 |