1998 Fiscal Year Final Research Report Summary
The multiplicative subgroups of division rings
| Project/Area Number |
09640080
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| Research Category |
Grant-in-Aid for Scientific Research (C)
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| Allocation Type | Single-year Grants |
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| Section | 一般 |
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| Research Field |
Algebra
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| Research Institution | Niihama National College of Technology |
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Principal Investigator |
CHIBA Katsuo Nlihama National College of Technolgy, Emgineering Science, Associate professor, 数理科, 助教授 (60141933)
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| Project Period (FY) |
1997 – 1998
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| Keywords | division ring / free group ring / skew polynomial ring / valuation / free field |
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| Research Abstract |
In [J.Algebra 185 (1996), 298-313], Figueiredo, Goncalves and Shirvani stated the following : The division ring of fractions of a skew polynomial ring over a rational function field of a single variable contains a non-commutative free gruop algebra, if it is not a P1-ring, It is difficalt to understand the proof of their results completely. They used the valuation of the division rings In this research we study free fields in division rings with complete discrete valuation. The main results are the followings.
1. Let D be a division ring with a complete discrete valuation and the center of D is infinite, if the dimention of D is infinite, then D contains the free field K <z (x, y)> on a set {x, y}. In particular, the multiplicative group of D contains a noncommutative free group.
2. For any commutative field k, the free field k <(x, y)> on a set {x, y} has discrete valuation v such that v(x)=m, v(x) n for any integers m, n.
It is interesting to cassify the discrete valations of the free field k <(x, y)> on k.
The following cojecture is more plausible. The division ring of fractions of the Weylalgebra over a field k of characteristic zero does not contain the free field k <(x, y)> on a set {x, y}
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