You were right the first time. If h=1 mod 4, the ring is Z[(1 + sqrt(h))/2]. Otherwise, the ring is Z[sqrt(h)]. It is understood that square factors are removed from h, so that h is squarefree. This is all derived in Harvey Cohn, "Advanced Number Theory", Dover, p. 45, a wonderful book. -- Gene From: Dan Asimov <asimov@msri.org> To: math-fun <math-fun@mailman.xmission.com> Sent: Friday, November 18, 2016 11:33 AM Subject: Re: [math-fun] The Hippasus Integers Correction:
On Nov 18, 2016, at 10:30 AM, Dan Asimov <asimov@msri.org> wrote:
Let O_h be the ring of algebraic integers of ℚ(√-h), Then:
O_1 = Z[i]
O_2 = Z[sqrt(-2)]
O_h = Z[(1 + sqrt(h))/2] for h = -3, -7, -11, -19, -43, -67, -163.
The last line should read: ----- O_h = Z[(1 + sqrt(h))/2] for -h = -3, -7, -11, -19, -43, -67, -163. ----- (and in that e-mail "discover" should be "discoverer". Argggh. —Dan