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Completely stuck on this question

Let $R = \mathbb{Z}[\sqrt{2}]$.

(i) Show that $u = 1 −\sqrt{2}$ is a unit in $R$.

(ii) Show that all elements of the form $\pm u^n$ for $n \in \mathbb{Z}$ are distinct units (i.e., show that each one is a unit and that no two of them are equal).

I've shown $u$ is a unit however cannot do the second part Thanks!

user26857
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B.tom
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1 Answers1

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HINT: $(1-\sqrt{2})(-1-\sqrt{2}) = 1$

Also $\pm(1-\sqrt{2})^n \cdot \pm (-1-\sqrt{2})^n = 1$

Too see that they are not equal to each other note that every number is a power of $u \not= \pm 1,0$

Stefan4024
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