Can you give me a really complete proof tha in a quadratic fiield $Q[\sqrt{(d)}]$ the algebraic integers are:
$\mathbb Z[\sqrt{(d)}]$ if $d\equiv2,3 \pmod4$
$\mathbb Z[(1+\sqrt{(d)})/2]$ if $d\equiv1 \pmod4$
Thank you very much :)
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Is the Galois theory available? Or are you seeking for an elementary proof? – awllower Feb 17 '14 at 10:43
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6See for example the answers here. This is also many other places on the internet (and in books). – anon Feb 17 '14 at 10:44
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3Dear Bar, One thing to note is that it is important in your statement that you take $d$ to be a square free integer. Regards, – Matt E Feb 17 '14 at 11:45
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Check Number Fields by Daniel A. Marcus. If I remember correctly, there is proof of this in that book. – Shreya May 17 '18 at 11:51
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There is a very complete proof in Dummit and Foote's text. – Oiler Apr 28 '20 at 18:38