ILL 1972 Problem 19

Let $S$ be a subset of the real numbers with the following properties: $(i)$ If $x \in S$ and $y \in S$ , then $x - y \in S$ ; $(ii)$ If $x \in S$ and $y \in S$ , then $xy \in S$ ; $(iii)$ $S$ contains an exceptional number $x'$ such that there is no number $y$ in $S$ satisfying $x'y + x' + y = 0$ ; $(iv)$ If $x \in S$ and $x \neq x'$ , there is a number $y$ in $S$ such that $xy+x+y = 0$ . Show that $(a)$ $S$ has more than one number in it; $(b)$ $x' \neq -1$ leads to a contradiction; $(c)$ $x \in S$ and $x \neq 0$ implies $1/x \in S$ .