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College Algebra Exam Review 266

College Algebra Exam Review 266 - 276 6 RINGS Example...

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276 6. RINGS Example 6.2.17. The kernel of the ring homomorphism KOExŁ ! K given by p 7! p.a/ is the set of all polynomials p having a as a root. Example 6.2.18. The kernel of the ring homomorphism C. R / ! C.S/ given by f 7! f j S is the set of all continuous functions whose restriction to S is zero. Example 6.2.19. Let K be a field. Define a map ' from KOExŁ to Fun .K; K/ , the ring of K –valued functions on K by '.p/.a/ D p.a/ . (That is, '.p/ is the polynomial function on K corresponding to the poly- nomial p .) Then ' is a ring homomorphism. The homomorphism property of ' follows from the homomorphism property of p 7! p.a/ for a 2 K . Thus '.p C q/.a/ D .p C q/.a/ D p.a/ C q.a/ D '.p/.a/ C '.q/.a/ D .'.p/ C '.q//.a/ , and similarly for multiplication. The kernel of ' is the set of polynomials p such that p.a/ D 0 for all a 2 K . If K is infinite, then the kernel is f 0 g , since no nonzero polynomial with coefficients in a field has infinitely many roots. If K is finite, then ' is never injective. That is, there always exist nonzero polynomials
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