A question about Z[sqrt(-p)]
Leonid Lenov
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.
Arturo Magidin
Yes; the only units in Z[sqrt(-p)] are 1 and -1, so 2 cannot be an
associate of either (1-sqrt(-p)) or (1+sqrt(-p)). A norm argument
shows that 2 is irreducible, and clearly 2 does not divide either 1-
sqrt(-p) nor 1+sqrt(-p).
Keep in mind, though, that Z[sqrt(-p)] may not be a number field: if
p=3 (mod 4), then the number ring of Q(sqrt(-p)) is Z[(1+sqrt(-p))/2].
So, for example, the number ring of Q(sqrt(-3)) *is* a UFD; that ring
is Z[(1+sqrt(-3))/2]. That said, there are only finitely many number
rings of imaginary quadratic number fields that are UFDs.
--
Arturo Magidin
Bill Dubuque
> On Jan 22, 9:48 am, Leonid Lenov <leonidle...@gmail.com> wrote:
>>
>> Let p>2 be a prime. In Z[sqrt(-p)] the following holds:
>> so there is no unique factorization in Z[sqrt(-p)]. Is that correct?
>
> Yes; the only units in Z[sqrt(-p)] are 1 and -1, so 2 cannot be an
> associate of either (1-sqrt(-p)) or (1+sqrt(-p)).
That's not needed. It suffices to show 2 is non-prime irreducible.
Norms => irreducible, and the factorization shows its non-prime.
> A norm argument shows that 2 is irreducible, and clearly 2 does not
> divide either 1-sqrt(-p) nor 1+sqrt(-p).
Achava Nakhash, the Loving Snake
In order to properly understand the other replies, here is some
information you should know. First of all, when you are in the ring Z
[sqrt(-p)], prime means prime in this ring, and being a prime among
the set of integers does not make you a prime in this ring. For
instance,
p = -(sqrt(-p)))^2, and so p is not a prime in this ring.
Second of all, unique factorization is always up to unit factors. A
unit is defined as an element u of the ring such that there exists an
element v of the ring with u*v = 1. In other words, the units are the
multiplicatively invertible elements of the ring. With the integers,
the only units are 1 and -1, so there is no real issue. In your ring Z
[sqrt(-p)], it looks like there are no units except 1 and -1.
However, if you had used Z{sqrt(p)] instead. there would have been
infinitely many units.
Third of all, number theorists don't generally consider the rings Z
[sqrt(-p)]. Instead they consider all the elements of Q[sqrt(-p)]
which satisfy monic polynomials over Z. In your cases, this is the
same as Z[sqrt(-p)] as long as p = 1 (mod 4). However, when p = 3
(mod 4), then instead of Z[sqrt(-p)] you get everything of the form (a
+ b*sqrt(-p))/2 where a and b are integers that are either both even
or both odd. This contains, but is larger than Z[sqrt(-p)]. When p =
3, you actually get units doing this when a = b = 1. For other p
there are no units.
You can never get unique factorization out of Z[sqrt(-p)] when p = 3
(mod 4). When you do it the standard way that I described above, you
get unique factorization for precisely 8 values of p. Unfortunately I
won't have access to a number theory book to look up which one's they
are until I get home from work which wll be maybe 4 hours from now.
However, I know that 3 is among them. In fact I think that they are
all = 3 (mod 4). You can adjoin sqrt(-1) also and that gives you a
unique factorization domain whose units are 1, -1, i, -i. You will
hear in some quarters about the 9 numbers of Gauss which include -1
and -p for all the different p that give you unique factorization
domains doing it the way I did. I am pretty sure that
-19 and -163 are among these numbers.
Regards,
Achava
Andrew Usher
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
Achava Nakhash, the Loving Snake
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
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