Let $F$ be a field and $R$ a subring of $F$ that is not a field. Let $x$ be a variable. Let $S = \left\{ a _ { 0 } + a _ { 1 } x + \cdots + a _ { n } x ^ { n } \mid n \geq 0 \text{ and } a _ { 0 } \in R , a _ { 1 } , \cdots a _ { n } \in F \right\}$. (A) (2 marks) Show that, with the natural operations of addition and multiplication of polynomials, $S$ is an integral domain. (B) (4 marks) Let $I = \{ f ( x ) \in S \mid f ( 0 ) = 0 \}$. Determine whether $I$ is a prime ideal. (C) (4 marks) Determine whether $S$ is a PID.
Let $F$ be a field and $R$ a subring of $F$ that is not a field. Let $x$ be a variable. Let $S = \left\{ a _ { 0 } + a _ { 1 } x + \cdots + a _ { n } x ^ { n } \mid n \geq 0 \text{ and } a _ { 0 } \in R , a _ { 1 } , \cdots a _ { n } \in F \right\}$.\\
(A) (2 marks) Show that, with the natural operations of addition and multiplication of polynomials, $S$ is an integral domain.\\
(B) (4 marks) Let $I = \{ f ( x ) \in S \mid f ( 0 ) = 0 \}$. Determine whether $I$ is a prime ideal.\\
(C) (4 marks) Determine whether $S$ is a PID.