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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.

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