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**sfspitfire23** I'm having trouble showing how $\displaystyle \alpha(1/4)=a^2\Rightarrow 1/4=a^2$. How would I do this?

We must show that $\displaystyle \mathbb{Q}$ and $\displaystyle \mathbb{Z}$ are not isomorphic. Consider the mapping $\displaystyle \alpha : \mathbb{Q} \rightarrow \mathbb{Z}$. Then we have $\displaystyle \alpha(1/2)=a$. Then we have $\displaystyle \alpha(1/2)^2=a^2$ which can be written as $\displaystyle \alpha((1/2)^2)=a^2$. So, $\displaystyle \alpha(1/4)=a^2$. We can pull this apart to get $\displaystyle \alpha(1/8)+\alpha(1/8)=a^2$. to yield $\displaystyle 1/4=a^2$. But there is nothing in $\displaystyle \mathbb{Z}$ that squares to $\displaystyle 1/4$. Therefore, they are not isomorphic.