Let $K$ be a number field and $\mathcal{O}_K$ be its ring of integers with unit rank $\geq 1$. For a given prime ideal $\mathfrak{p}$, shall we say that the map $\mathcal{O}_K^\times \to (\mathcal{O}_K/\mathfrak{p})^\times$ is surjective, where $\mathcal{O}_K^\times$ is the unit group.
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$\begingroup$ Have you tried it with $K= \mathbb{Q}$? $\endgroup$– user136098Commented Jul 7, 2022 at 14:38
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$\begingroup$ I am taking number fields $K$ with unit rank at least one. $\endgroup$– KannanCommented Jul 7, 2022 at 14:55
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$\begingroup$ This is not always true, see e.g. Chen, Kitaoka, Yu - Distribution of units of real quadratic number fields - 2000. $\endgroup$– WatsonCommented Jul 7, 2022 at 15:00
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1$\begingroup$ You could also look at the fundamental units, $\epsilon_D$, of real quadratic fields and find one with a prime $p\ge 5$ such that $p| \mathrm{Norm}_{\mathbb{Q}(\sqrt{D})/\mathbb{Q}}(\epsilon_D-1) $. $\endgroup$– user136098Commented Jul 7, 2022 at 15:07
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2$\begingroup$ See also mathoverflow.net/questions/141886 $\endgroup$– WatsonCommented Jul 7, 2022 at 16:15
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