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I'm asked to show that $x=2+\sqrt{2}$ is irreducible in $\mathbb{Z}[\sqrt{2}]$ by using the norm map $$N:\mathbb{Z}[\sqrt{2}]\rightarrow \mathbb{Z}^+:a+\sqrt{2}b\mapsto |a^2-2b^2|$$

Now, if $x=yz$, then $2=N(x)=N(y)N(z)$ forcing wlog $N(y)=1$. I'm now stuck trying to show that $y$ must be a unit and would appreciate any help.

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Use the definition of the norm. If $y=c+d\sqrt{2}$ then $N(y)=(c+d\sqrt{2})(c-d\sqrt{2})=1$.

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Write $y=u+\sqrt2 v$, $N(y)=u^2-2v^2=(u+\sqrt 2 v)(u-\sqrt2 v)=1$ or $-1$ implies that $u-\sqrt 2v$ or $-(u-\sqrt2 v)$ is the inverse of $y$.

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