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| Main Author: | |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2506.00266 |
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| _version_ | 1866912853765652480 |
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| author | Hofmann, Tommy |
| author_facet | Hofmann, Tommy |
| contents | We consider the computational problem of determining the unit group of a finite ring, by which we mean the computation of a finite presentation together with an algorithm to express units as words in the generators. We show that the problem is equivalent to the number theoretic problems of factoring integers and solving discrete logarithms in finite fields. A similar equivalence is shown for the problem of determining the abelianization of the unit group or the first $K$-group of finite rings. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_00266 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Determining unit groups and $\mathrm{K}_1$ of finite rings Hofmann, Tommy Number Theory Symbolic Computation K-Theory and Homology Rings and Algebras Primary 68W30, 19D50, 20C05, 19-08, 11Y16, Secondary 16Z05 We consider the computational problem of determining the unit group of a finite ring, by which we mean the computation of a finite presentation together with an algorithm to express units as words in the generators. We show that the problem is equivalent to the number theoretic problems of factoring integers and solving discrete logarithms in finite fields. A similar equivalence is shown for the problem of determining the abelianization of the unit group or the first $K$-group of finite rings. |
| title | Determining unit groups and $\mathrm{K}_1$ of finite rings |
| topic | Number Theory Symbolic Computation K-Theory and Homology Rings and Algebras Primary 68W30, 19D50, 20C05, 19-08, 11Y16, Secondary 16Z05 |
| url | https://arxiv.org/abs/2506.00266 |