Separable Algebras and Galois Theory |
Common terms and phrases
a₁ algebraic closure assertion Aut F Aut(F automorphism b₁ b₂ basis closure of F commutative ring conditions are equivalent defined denote direct sum direct summand easily seen element separable F into F F is normal f is separable field extension finite-dimensional following diagram commute Galois theory group map Home-Alg F,E HomE-Mod H,T Homset hypothesis idempotent imbedding independence of characters induced inner derivation irreducible polynomial isomorphism left ideal left TOP-module linear factors magic element 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 ring map root separable by Theorem separable extension separable polynomials separable R-algebra splits into linear splitting field subalgebra subfield subgroup surjective tensor product U U°P-module U-U bimodule unique E-algebra map well-defined whence F Σα