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Can we show that the ring of Gaussian integers $$\mathbb{Z}[\sqrt{17}]:=\{a+b\sqrt{17}:a,b\in\mathbb{Z}\}$$ $$\mathbb{Z}[\sqrt{11}]:=\{a+b\sqrt{11}:a,b\in\mathbb{Z}\}$$

equipped with standard addition and multiplication are not isomorphic?

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If they were isomorphic, then wouldn't they both have the same number of solutions of the equation $x^2=11$? – Old John Oct 27 '13 at 18:16
up vote 3 down vote accepted

An isomorphism of rings would map $1$ to $1$. Hence it would fix all integers. In $\mathbb Z[\sqrt{17}]$ we have $(\sqrt{17})^2 = 17$. It is easy to check that no element $x$ of $\mathbb Z[\sqrt{11}]$ satisfies $x^2 = 17$.

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Hint: Both rings can be equipped with multiplicative norms. Use this to show that $11$ is the product of nontrivial divisors in one ring, but not the other.

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