# why $\mathbb Z \ncong \mathbb Z$?

The following is a question from section $3.11$ of the book An introduction to abstract algebra by Allenby:

Explain intuitively why $\mathbb Z[\sqrt 2] \ncong \mathbb Z[\sqrt 3]$.back your intuition with proof.

Note:this example not only says that $\theta: a+b\sqrt 2 \mapsto a+b\sqrt 3$ is not isomorphism .It says no isomorphism can be found at all – no matter how clever choice of mapping you try to make ..

I can’t see what’s the intution behind this ..can anyone provide some hint on this…