Galois groups acting as linear fractional transformations
نویسندگان
چکیده
منابع مشابه
Linear algebraic groups as parameterized Picard–Vessiot Galois groups
We show that a linear algebraic group is the Galois group of a parameterized Picard-Vessiot extension of k(x), x′ = 1, for certain differential fields k, if and only if its identity component has no one dimensional quotient as a linear algebraic group.
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Writing f(T ) = (T − r1) · · · (T − rn), the splitting field of f(T ) over K is K(r1, . . . , rn). Each σ in the Galois group of f(T ) over K permutes the ri’s since σ fixes K and therefore f(r) = 0⇒ f(σ(r)) = 0. The automorphism σ is completely determined by its permutation of the ri’s since the ri’s generate the splitting field over K. A permutation of the ri’s can be viewed as a permutation ...
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Writing f(X) = (X− r1) · · · (X− rn), the splitting field of f(X) over K is K(r1, . . . , rn). Each σ in the Galois group of f(X) over K permutes the ri’s since σ fixes K and therefore f(r) = 0⇒ f(σ(r)) = 0. The automorphism σ is completely determined by its permutation of the ri’s since the ri’s generate the splitting field over K. A permutation of the ri’s can be viewed as a permutation of th...
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ژورنال
عنوان ژورنال: Journal of Algebra
سال: 1989
ISSN: 0021-8693
DOI: 10.1016/0021-8693(89)90034-3