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

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You really want to require that $d$ be positive, no? –  Dylan Moreland Mar 21 '12 at 13:37
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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
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1 Answer

up vote 4 down vote accepted

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
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