Upper bounds for cyclotomic numbers
Let q be a power of a prime p, let k be a nontrivial divisor of q−1 and write e=(q−1)/k. We study upper bounds for cyclotomic numbers (a,b) of order e over the finite field Fq. A general result of our study is that (a,b)≤3 for all a,b∈Z if p>(14−−√)k/ordk(p). More conclusive results will be obtai...
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| Main Authors: | , , |
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| Other Authors: | |
| Format: | Article |
| Language: | English |
| Published: |
2020
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| Subjects: | |
| Online Access: | https://hdl.handle.net/10356/145013 |
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| Institution: | Nanyang Technological University |
| Language: | English |
| Summary: | Let q be a power of a prime p, let k be a nontrivial divisor of q−1 and write e=(q−1)/k. We study upper bounds for cyclotomic numbers (a,b) of order e over the finite field Fq. A general result of our study is that (a,b)≤3 for all a,b∈Z if p>(14−−√)k/ordk(p). More conclusive results will be obtained through separate investigation of the five types of cyclotomic numbers: (0,0),(0,a),(a,0),(a,a) and (a,b), where a≠b and a,b∈{1,...,e−1}. The main idea we use is to transform equations over Fq into equations over the field of complex numbers on which we have more information. A major tool for the improvements we obtain over known results is new upper bounds on the norm of cyclotomic integers. |
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