| Budget Amount *help |
¥1,200,000 (Direct Cost: ¥1,200,000)
Fiscal Year 1998: ¥600,000 (Direct Cost: ¥600,000)
Fiscal Year 1997: ¥600,000 (Direct Cost: ¥600,000)
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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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