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a e F a e F\E A®RB algebraic closure algebraic extension assertion AutgF automorphism basis closure of F commutative ring conditions are equivalent defined denote direct summand E-algebra map easily seen element separable F E F F into F F is normal field extension finite-dimensional following diagram commute Galois theory group map HomSet(S,F hypothesis imbedding independence of characters induced inner derivation irreducible polynomial isomorphism left A-module left ideal left Uop-module linear factors M®RN magic element maximal ideal minimal length morphisms multiplication map nonzero element normal closure normal extension preceding theorem projective left projective modules PROOF pullback R-algebra isomorphism R-linear R-module map right T-module ring map root separable by Theorem separable extension separable polynomials separable R-algebra splits into linear splitting field subalgebra subfield Suppose f surjective tensor product U-U bimodule u®lop)z U®Uop unique R-algebra map Uop-module map well-defined whence F