Chapter 6, Problem 38E

Let $G=\left\{a+b\sqrt{2}|a,b\text{are rational}\right\}$ and $H=\left\{\begin{array}{l}a2b\\ ba\end{array}|a,b\text{are rational}\right\}$ .Show that G and H are isomorphic under addition. Prove that G andH are closed under multiplication. Does your isomorphism preservemultiplication as well as addition? (G and H are examples of rings—a topic we will take up in Part 3.)