Multiplicative relations with conjugate algebraic numbers
We study which algebraic numbers can be represented by a product of conjugate over a fixed number field K algebraic numbers in fixed integer powers. The problem is nontrivial if the sum of these integer powers is equal to zero. The norm over K of such number must be a root of unity. We show that t...
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| Date: | 2007 |
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| Main Author: | |
| Format: | Article |
| Language: | English |
| Published: |
Інститут математики НАН України
2007
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| Series: | Український математичний журнал |
| Subjects: | |
| Online Access: | https://nasplib.isofts.kiev.ua/handle/123456789/164212 |
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| Journal Title: | Digital Library of Periodicals of National Academy of Sciences of Ukraine |
| Cite this: | Multiplicative relations with conjugate algebraic numbers / A. Dubickas // Український математичний журнал. — 2007. — Т. 59, № 7. — С. 890–900. — Бібліогр.: 20 назв. — англ. |
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Digital Library of Periodicals of National Academy of Sciences of Ukraine| Summary: | We study which algebraic numbers can be represented by a product of conjugate over a fixed number
field K algebraic numbers in fixed integer powers. The problem is nontrivial if the sum of these integer
powers is equal to zero. The norm over K of such number must be a root of unity. We show that there
are infinitely many algebraic numbers whose norm over K is a root of unity and which cannot be
represented by such product. Conversely, every algebraic number can be expressed by every sufficiently
long product in conjugate over K algebraic numbers. We also construct nonsymmetric algebraic
numbers, i.e., such that none elements of the respective Galois group acting on the full set of their
conjugates form a Latin square. |
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