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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/2504.10202 |
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| _version_ | 1866916763840544768 |
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| author | Kalmynin, Alexander |
| author_facet | Kalmynin, Alexander |
| contents | In this paper, we employ a version of Stepanov's method, developed by Hanson and Petridis, to prove several results on additive irreducibility of multiplicative subgroups in finite fields of prime order $p$. Specifically, we show that if a subgroup $μ_d$ of $d$-th roots of unity satisfies $A-A=μ_d\cup\{0\}$, then $d=2$ or $6$. Additionally, we resolve the Sárközy's conjecture on quadratic residues: for prime $p$, the set $\mathcal R_p$ of quadratic residues modulo $p$ cannot be represented as $A+B$ for $A,B$ with $\min(|A|,|B|)>1$. More generally, we prove that if the set of $d$-th roots of unity $μ_d$ is represented non-trivially as $A+B$, then the sizes of summands are equal. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_10202 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On additive irreducibility of multiplicative subgroups Kalmynin, Alexander Number Theory In this paper, we employ a version of Stepanov's method, developed by Hanson and Petridis, to prove several results on additive irreducibility of multiplicative subgroups in finite fields of prime order $p$. Specifically, we show that if a subgroup $μ_d$ of $d$-th roots of unity satisfies $A-A=μ_d\cup\{0\}$, then $d=2$ or $6$. Additionally, we resolve the Sárközy's conjecture on quadratic residues: for prime $p$, the set $\mathcal R_p$ of quadratic residues modulo $p$ cannot be represented as $A+B$ for $A,B$ with $\min(|A|,|B|)>1$. More generally, we prove that if the set of $d$-th roots of unity $μ_d$ is represented non-trivially as $A+B$, then the sizes of summands are equal. |
| title | On additive irreducibility of multiplicative subgroups |
| topic | Number Theory |
| url | https://arxiv.org/abs/2504.10202 |