I am aware that $\sqrt{-5}$ is both prime and irreducible in $\mathbb Z [\sqrt{-5}]$ . And that $2, 3 $ etc., are irreducible but not prime. My question is are there other prime elements in $\mathbb Z [\sqrt{-5}]$?
Thanks for the response.
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4Integer primes $p\equiv 11,13,17,19\pmod {20}$ are prime here as well. These are the primes where $-5$ is not a quadratic residue. – Thomas Andrews Jul 22 '23 at 09:13
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2Others are $6+\sqrt{-5},$ since it's norm is $41.$ – Thomas Andrews Jul 22 '23 at 10:29
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Why do you say "both prime and irreducible"? https://math.stackexchange.com/questions/405759 Besides, Quick beginner guide for asking a well-received question + please avoid "no clue" questions (mainly: show your attempts to solve your question). – Anne Bauval Jul 22 '23 at 10:36
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Acknowledged, will follow the guide – Phalaksha C G Jul 22 '23 at 11:31
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Sorry I'm unable to understand what $\oplus$ stands for. – Phalaksha C G Jul 22 '23 at 11:35
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@ThomasAndrews can you please elaborate? – Phalaksha C G Jul 22 '23 at 11:35
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No, because you haven't told me what confuses you in my comment. Be specific. @PhalakshaCG – Thomas Andrews Jul 22 '23 at 16:30
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Why have you taken 20 in the mod. And does "-5 is not a quadratic residue" mean -5 not ≡ n^2 mod p ? – Phalaksha C G Jul 23 '23 at 13:14