Mueller Quade, Steinwandt. Основные алгоритмы для рациональных областей функции , Mueller-Quade, Steinwandt. Basic algorithms for rational function fields (JSC1999)(T)(28s).djvu:

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Mueller Quade, Steinwandt. Основные алгоритмы для рациональных областей функции

Mueller-Quade, Steinwandt. Basic algorithms for rational function fields (JSC1999)(T)(28s).djvu

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Date Dec 19, 2004

Cites: ("type of a field extension"): Given K(g) < K(x) find the transcendence
degree t o/K(x) over K(g)...
гтЦ !/ i~ ^tj, Z i~ J,(/ i~ (/ Z i~ ^t(/Z 5
xby3z 4- 4ж5уг3 4- 4ж3у52 4- x3yzb 4- xybz3 4- 4жу325
4ж62 4- ж5у2 4- Ж225 4- 4жу6 4- 4y5z2 4- yz6 3
Ж5у32 _)_ 4x5yz3 + Ax3y5z + x3yzb 4- жу523 4- 4жу325
Ж623 4- 4ж5у4 4- 4ж425 4- ж3у6 4- ?/524 4- 4y3z6
Ж5у32 _)_ 4x5yz3 + Ax3y5z + x3yzb 4- жу523 4- 4жу325 '
Z6+Dx2 4- 4w2 4- 4:Z2)Z4 4- fa;27j2 4- ж222 4- w2z2)Z2 4- 4a;2w2z2
From this we immediately recognize ?^(x,y,z)/?^(x,y,z)z^z as a separable algebraic
extension of degree 6...
gGB
As deg(g) = deg(ip(g)) for g € В we have [K(x) : K(B)} < Цдеё deg(g).a
We can now solve
Problem 5...
Then for K(x)/K(g) algebraic the following statements hold:
(i) J/K(x)/L is separable then there is А С {1,...,/}; After removing the elements
of К the coefficients of a reduced Grobner basis of Плел ^*л form the canonical
generating set o/L as computed by Algorithm 1.8...
it consists
of 184 elements, and its field of invariants over <Ц>(С2з) is generated by
Х132Х2Ы + С239Ж114Ж232,Ж130Ж216 + С237Ж116Ж230,Ж128Ж218 + СгЗ^!18^28,
X224:,Xi92 + Ж292}...