Question

Exercise 2.109.1 Mimic Example 2.97 and construct a homomorphism from Rx to C that sends p(x) to p(i) and prove that it is surjective with kernel (2+1). Then apply Theorem 2.107 to establish the claim that R[C]/(x +1) C.

Answer 1

Ex: xet us define a mapping di IR [u] C by & (P(x)) = P(i) where P(x) E IR [4] - K + P (3), (*) ( IP TP. Now, $ CP(X) + 9(x)) = P(i) + 9(i) - $ CP(x)) + Ø 69()) And , o (P(x) q (x)) - pli) & (i) àd (P(x)) & C9 (x)). 0 is homomorphism Now, we show that Ø is sunjective. det, n(i) te then there is a pollynomial m(x) in IR to such that Ø (m(x)) = n(i). so o is subjective. Hence, I is epimorphism. Then, by fundamental Theorem of Homomorphism. of Rings we have, IR [x] /kend řce

Now, ker & a {p(x) E IR [x] : 0 (0 (2)) 20} as p(x) E 1R EN] : P(i)=0} as p(x) E IR (A) : (x+4) | f(x)} a kartly Therefore, IR [a] / Kartly Fc.