By R. Wait
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Additional info for Numerical solution of algebraic equations
Example text
An algebraic group that does not admit any linear representation is called quasiabelian. In other words, a quasi-abelian variety is an algebraic group G such that OG (G) = C. Algebraic groups over an algebraic closed base field C, which are complete and connected, are called abelian varieties. Since they are complete varieties, they do not admit non-constant global regular functions and then they are quasi-abelian. The following results give us the structure of the algebraic groups by terms of linear and quasi-abelian algebraic groups.
Then there is σ ∈ G(C) such that x · σ = y, and by the commutativity of the diagram we have s · σ = y. 27. Let x and y be two closed points of Diff(GK , ∂A ). Then there exists an invertible K-isomorphism of differential fields κ(x) κ(y). Proof. There is a closed point σ ∈ G, such that x · σ = y. Then Rσ : (GK , ∂A ) → (GK , ∂A ) is an automorphism that maps x to y. Then it induces an invertible K-isomorphism Rσ : κ(y) → κ(x). 28. For each closed point x ∈ Diff(GK , ∂A ) we say that the differential extension K ⊂ κ(x) is a Galois extension associated to the non-autonomous differential algebraic dynamical system (GK , ∂A ).
Any Galois extension associated to this last equation is K1 -isomorphic to L. By the induction hypothesis the extension K1 ⊂ L is a Kolchin extension, hence K ⊂ L is a Kolchin extension. 21. Assume that G is affine and solvable. Then K ⊂ L is a strictLiouville extension. Proof. The Galois group is a subgroup of G, and then it is a solvable group. 20. 22. If there is a connected affine solvable group H ⊂ G such that Galx (GK , ∂A ) ⊂ H, then K ⊂ L is a strict-Liouville extension. Proof. H is connected affine solvable an then it has trivial Galois cohomology.