(A) (5 marks) Let $X$ be a non-empty finite set and let $R$ be the ring of $\mathbb { Z }$-valued functions on $X$, with pointwise addition and multiplication. Let $S$ be an additive subgroup of $R$ such that the multiplicative identity $1 _ { R } \notin S$ and such that for all $s , s ^ { \prime } \in S , s s ^ { \prime } \in S$. Show that there exists $x \in X$ and a prime number $p$ such that $f ( x )$ is divisible by $p$ for all $f \in S$ (Hint: Consider the sets $\{ f ( x ) : f \in S \}$ for all $x \in X$.)
(B) (5 marks) Let $K$ be a subfield of $\mathbb { C }$ with $[ K : \mathbb { Q } ] = 2$. Let $P \in \mathbb { Q } [ x ]$ be irreducible over $\mathbb { Q }$. Show that $P$ is either irreducible in $K [ x ]$ or splits as product of two irreducible polynomials in $K [ x ]$.