Linear relations among radicals
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909886983438336 |
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| author | Perucca, Antonella |
| author_facet | Perucca, Antonella |
| contents | Let $K$ be a field, fix an algebraic closure $\overline{K}$, and let $G$ be a subgroup of $\overline{K}^\times$. We are able to give a closed formula for the ratio between the degree $[K(G):K]$ and the index $|GK^\times:K^\times|$, provided that the latter is finite. Our formula explains all the $K$-linear relations among radicals, which (beyond the ones stemming from the multiplicative group $GK^\times/K^\times$) are generated by relations among roots of unity and single radicals. Our work builds on results by Rybowicz, which in turn are based on work by Kneser and Schinzel. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_02498 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Linear relations among radicals Perucca, Antonella Number Theory 12J99, 11R18 Let $K$ be a field, fix an algebraic closure $\overline{K}$, and let $G$ be a subgroup of $\overline{K}^\times$. We are able to give a closed formula for the ratio between the degree $[K(G):K]$ and the index $|GK^\times:K^\times|$, provided that the latter is finite. Our formula explains all the $K$-linear relations among radicals, which (beyond the ones stemming from the multiplicative group $GK^\times/K^\times$) are generated by relations among roots of unity and single radicals. Our work builds on results by Rybowicz, which in turn are based on work by Kneser and Schinzel. |
| title | Linear relations among radicals |
| topic | Number Theory 12J99, 11R18 |
| url | https://arxiv.org/abs/2511.02498 |