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I want to prove that the ring of integers $O_K$ in a real quadratic field $K$ contains a non-trivial unit. I'm given the hint that I should use a theorem concerning lattices and convex sets. This seems pretty clearly to be pointing to Minkowski's theorem on lattices. But I don't really see how that can be applied to finding units.

Elie Bergman
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  • See https://math.stackexchange.com/a/1220631/589 for instance – lhf May 24 '18 at 16:32
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    I am interested in a geometric proof, not one using continued fractions. – Elie Bergman May 24 '18 at 16:33
  • Minkowski's theorem is the cornerstone of the proof of Dirichlet's theorem which gives the $\mathbf Z$-rank of the group of units of a number field. Perhaps your exercise aims simply to adapt this proof to show that the $\mathbf Z$-rank of the units is $1$ for a real quadratic field. This should be easier than the general case thanks to a simpler description of the compact convex symmetrical body to which you apply Minkowski's theorem. – nguyen quang do May 26 '18 at 06:26

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