§ Quotient by maximal ideal gives a field
§ Quick proof
Use correspondence theorem. only has the images of as ideals which
is the zero ideal and the full field.
§ Element based proof
Let be an element in . Since , we have .
Consider . By maximality of , . Hence there exist elements
such that . Modulo , this read m. Thus
is an inverse to , hence every nonzero element is invertible.