Consider the following statement: Let $F$ be a field and $R = F [ X ]$ the polynomial ring over $F$ in one variable. Let $I _ { 1 }$ and $I _ { 2 }$ be maximal ideals of $R$ such that the fields $R / I _ { 1 } \simeq R / I _ { 2 } \neq F$. Then $I _ { 1 } = I _ { 2 }$.
Prove or find a counterexample to the following claims:\\
(A) The above statement holds if $F$ is a finite field.\\
(B) The above statement holds if $F = \mathbb { R }$.