Answer to Question #146090 in Abstract Algebra for Sourav Mondal

Question

Show that( R [x] /<x²+1>) isomorphic To C as fields.

Answer 1

We give a map $f:\mathbb{R}[X]\longrightarrow \mathbb{C}.$ $x\mapsto i.$ In general, $g(x)\mapsto g(i).$ Hence $a+bX\mapsto a+ib.$ Hence surjective. Homomorphism is obvious since $gh(i)=g(i)h(i)$ for polynomials. Also $(g+h)(i)=g(i)+h(i).$ Now for kernel. $g(x)\mapsto 0\Rightarrow g(i)=0.$ Now by division algorithm, $g(x)=aX+b+(X^2+1)(q(x))$ . Hence $g(i)=0\Rightarrow ai+b=0\Rightarrow a=b=0.$ Hence

kernel is generated by $X^2+1.$ Hence by first isomorphism theorem, the result follows.