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Field of definition
Galois descent The case of schemes
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abelian affine algebraic k-set algebraic extension algebraic group defined algebraic k-set algebraic set defined assume automorphisms bijective closed subgroup closed subset commutative group composition series connected solvable connected unipotent contains Corollary Definition denote dense diagram dim G direct product elements of G exists fixed point follows function system G defined galois extension galois group group G group of exponent group scheme homomorphism induction hypothesis irreducible isomorphic k-algebra k-characters k-closed k-connected k-free k-group k-homomorphism k-isomorphism k-locally k-morphism k-rational k-regular k-split k-torus k-wound kernel Lemma Let f Let G Let H linearly disjoint locally closed mapping morphism morphism f Mult noetherian non-trivial open set polynomial pre-algebraic pre-algebraic k-set prescheme Proof Proposition regular functions resp solvable group Spec structure subgroup G subgroup of G Suppose Theorem topological space torus defined unipotent group unique upper triangular vector group vector group defined vector space