I consider the ideal $I \subseteq \Bbb Z$ with $I=\{n \in \Bbb Z \mid n \times \sqrt{2} \in \Bbb Z\}$. I do not manage to prove that this ideal is trivial. Can you help me ? Thanks.
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1Cf. this question – J. W. Tanner Jan 17 '21 at 18:42
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2Does this answer your question? Irrationality $\sqrt{2}$ with ideal – Pedro Amaral Jan 17 '21 at 18:44
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See also this post. – Dietrich Burde Jan 17 '21 at 19:36