This preview shows page 1. Sign up to view the full content.
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...
View
Full
Document
This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.
 Fall '08
 EVERAGE
 Algebra, Polynomials, Division, Remainder, Sets

Click to edit the document details