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Let $k$ be a real quadratic field and let $\epsilon$ be its fundamental unit.
Let $p$ be an odd prime number and denote $v_1$ and $v_2$ the primes above $p$ in the field $k$ ( we can suppose that there is just one ).
Is it there a known condition on the fundamental discriminant of $k$ such that $\epsilon$ is not a $p$-power in the completions $k_{v_1}$ and $k_{v_2}$ ?

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