The set $S\subseteq \mathbb{R}$ is given with the properties: $(a) \mathbb{Z}\subset S$, $(b) (\sqrt 2 +\sqrt 3)\in S$, $(c)$ If $x,y\in S$ then $x+y\in S$, and $(d)$ If $x,y\in S$ then $x\cdot y\in S$. Prove that $(\sqrt 2+\sqrt 3)^{-1}\in S$.
Source: Kosovo MO 2010 Grade 12, Problem 2
Tags: algebra
The set $S\subseteq \mathbb{R}$ is given with the properties: $(a) \mathbb{Z}\subset S$, $(b) (\sqrt 2 +\sqrt 3)\in S$, $(c)$ If $x,y\in S$ then $x+y\in S$, and $(d)$ If $x,y\in S$ then $x\cdot y\in S$. Prove that $(\sqrt 2+\sqrt 3)^{-1}\in S$.