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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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    You really want to require that $d$ be positive, no?2012-03-21
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    Have you covered Pell's equation?2012-03-21
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    If I recall correctly this can be shown using the norm function?2012-03-21
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    @Dylan: Thanks, I've edited2012-03-21

1 Answers 1

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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.2012-03-21