Answer to Question #138988 in Abstract Algebra for Sohan kumar

Question

Give two distinct maximal ideals in the polynomial ring Q[x] with justification.

Answer 1

Theorem:Let $F$ be a field and let $I$ be an ideal of the polynomial ring $F[x]$ ,then

$I$ is maximal if and only if $I=<p(x)>$ for some irreducible polynomial $p(x)\in F[x]$.

Example:

(1). Let $F=\Bbb Q$ and $p(x)=x^2+1$ , as here p(x) is irreducible in $\Bbb Q$ ,thus by above theorem $<p(x)>$ is a maximal ideal in $\Bbb Q[x]$ .

(2). Let field is same as (1) and $I=<q(x)=x^2+x+1>$ is maximal as same reason as in (1).