AAR6

Show that $(x-5)\,$ is a maximal ideal of $\mathbb{C}[x]\,$ .

If it could be shown that $\mathbb{C}[x]/(x-5) \cong \mathbb{C}\,$ then since $\mathbb{C}\,$ is a field, $(x-5)\,$ would be maximal.

Define $\phi:\mathbb{C}[x]\rightarrow\mathbb{C}\,$ by $f\mapsto f(5)\,$ .

Check that $\phi\,$ is an epimorphism and its kernel is $(x-5)\,$ .

Second method:

$\mathbb{C}[x]\,$ is a principal ideal domain. An ideal of a PID $(p)\,$ is maximal iff $p\,$ is irreducible in $\mathbb{C}[x]\,$ .

Since $x-5\,$ is irreducible, $(x-5)\,$ is maximal.