# Units in $\mathbb{Z}[\sqrt{d}]$ [duplicate]

Possible Duplicate:
infinitely many units in $\mathbb{Z}[\sqrt{d}]$ for any $d\gt1$.

This is an exercise of algebraic number theory.

Prove that in $\mathbb{Z}[\sqrt{d}] \$ , d square-free integer, $d > 0 \$, there are infinite many units.

Any hint ?

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## marked as duplicate by lhf, Dylan Moreland, Byron Schmuland, t.b., anonMar 21 '12 at 18:40

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

You really want to require that $d$ be positive, no? – Dylan Moreland Mar 21 '12 at 13:37
Have you covered Pell's equation? – Thomas Andrews Mar 21 '12 at 13:41
If I recall correctly this can be shown using the norm function? – sxd Mar 21 '12 at 13:41
@Dylan: Thanks, I've edited – WLOG Mar 21 '12 at 18:18

## 1 Answer

Here is a hint: The equation $$y^2-dx^2=1$$ has infinitely many solutions. See Pell's equation.

Another hint: What do we know about the norm of a unit? What happens if an element has norm $1$? Recall that the norm is multiplicative.

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The main point here is that it suffices to find one nontrivial unit, that is, a nontrivial solution of Pell's equation. – lhf Mar 21 '12 at 14:27