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From: Achava Nakhash, the Loving Snake on 24 Jan 2010 00:47
On Jan 22, 3:53 pm, Andrew Usher <k_over_hb...(a)yahoo.com> wrote:
> On Jan 22, 9:48 am, Leonid Lenov <leonidle...(a)gmail.com> wrote:
>
> > Hello,
> > Let p>2 be a prime. In Z[sqrt(-p)] the following holds:
> >    (1-sqrt(-p))(1+sqrt(-p))=1+p=2t
> > so there is no unique factorization in Z[sqrt(-p)]. Is that correct?
> > Thanks.
>
> Yes, and p need not be prime. As others have pointed out, if you allow
> the half-integer values when p=3 mod 4, you have additional UFDs: p=3,
> 7, 11, 19, 43, 67, 163.
>
> Andrew Usher

I just looked up the list of all quadratic fields which are obtained
by adjoining the square root of a negative number and which are unique
factorization domains. I found it in an interesting-looking paper
that Dorian Goldfeld wrote in 1985 which I would link if I knew how to
do it. The URL is

http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.bams/1183552617

The discfiminants of these fields are D = - 3 , - 4, - 7 , - 8 , -
11 , -19, -43, -67, -163.
From the OP's perspective, none of these are valid. From the number
field perspective, we are looking at the ring of algebraic integers in
the quadratic number fields generated by the square root of -1, -2,
-3, -7, -11, -19, -43, -67, -163.

It was shown by Theodore Motzkin around 1949 that Q[sqrt(-19))
produces a ring which is not Euclidean, not merely in the norm from
algebraic number theory, but in any possible norm and yet which is a
unique factorization domain. The actual paper was referenced in all
the modern algebra textbooks from my undergraduate days. I was amazed
around 1976 to meet Motzkin's son at a gathering at the apartment of a
girl in the next building over and so find out that he was a real
person and not just a name in textbooks.

Regards,
Achava
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