Unformatted text preview: Q ' W R=I ! S satisfying Q ' ı ± D ' . We have only to verify that Q ' also respects multiplication. But this follows at once from the deﬁnition of the product on R=I : Q '.a C I/.b C I/ D Q '.ab C I/ D '.ab/ D '.a/'.b/ D Q '.a C I/ Q '.b C I/: n Example 6.3.5. Deﬁne a homomorphism ' W R ŒxŁ ! C by evaluation of polynomials at i 2 C , '.g.x// D g.i/ . For example, '.x 3 ± 1/ D i 3 ± 1 D ± i ± 1 . This homomorphism is surjective because '.a C bx/ D a C bi . The kernel of ' consists of all polynomials g such that g.i/ D . The kernel contains at least the ideal .x 2 C 1/ D .x 2 C 1/ R ŒxŁ because i 2 C 1 D . On the other hand, if g 2 ker .'/ , write g.x/ D .x 2 C 1/q.x/ C...
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 Fall '08
 EVERAGE
 Algebra, Polynomials, Division, Remainder, Sets, Normal subgroup, Homomorphism, Epimorphism, isomorphism theorem, Homomorphism theorem

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