From: dean@ucdmath.ucdavis.edu (Dean Hickerson)
Newsgroups: sci.math.research
Subject: Euclidean rings (Re: Quadratic Fields)
Date: Sat, 17 Jun 1995 13:18:45 -0700
In article <1995Jun16.190331.22984@njitgw.njit.edu> katz@galois.njit.edu
(Robert Katz) asked:
>Consider the quadratic number fields, a + b * sqrt (d), where
>a, b are rational and d is a square free integer, positive or
>negative. For which d is the field a
>i) Euclidean domain
>ii) Principal ideal domain
>iii) unique factorization domain ?
And Noam Elkies (elkies@ramanujan.harvard.edu) replied:
> [The *field* is, of course, a UFD, but presumably you mean the ring
> of integers... Anyway, (ii) <-> (iii). For d<0 it's known that there
> are only 9 fields and 13 rings (four of which are proper subrings of
> the rings of integers) which have unique factorization. The Euclidean
> domains with both d<0 and d>0 are known, but I don't remember the list
> or a good reference. It's conjectured that there are infinitely many
> UFD's with d>0 but there's probably no proof coming soon. --NDE]
In a followup article, Bob Silverman (bobs@mathworks.com) said:
> You can find it in Hardy & Wright.
>
> The five Euclidean fields with d < 0 are -1, -2, -3, -7, -11.
> With d > 0 we have 2,3,5,6,7,11,13,17,19,21,29,33,37,41,57,73.
This is only partially correct. The general definition of a Euclidean
ring is that there is some function n from the nonzero elements of the
ring to the nonnegative integers for which the division algorithm works.
(Specifically, for any a and b in the ring with b nonzero, there exist
q and r in the ring with a=qb+r and either r=0 or n(r) katz@galois.njit.edu
(Robert Katz) asked:
|> Consider the quadratic number fields, a + b * sqrt (d), where
|> a, b are rational and d is a square free integer, positive or
|> negative. For which d is the field a
|> i) Euclidean domain
|> ii) Principal ideal domain
|> iii) unique factorization domain ?
To which Guest Moderator Noam Elkies (elkies@ramanujan.harvard.edu) added:
|> [The *field* is, of course, a UFD, but presumably you mean the ring
|> of integers... Anyway, (ii) <-> (iii). For d<0 it's known that there
|> are only 9 fields and 13 rings (four of which are proper subrings of
|> the rings of integers) which have unique factorization.
I wonder what you were thinking of here? `UFD' implies `integrally closed';
a proper subring of the ring of integers cannot have unique factorization.
In Z[sqrt(-3)] , e.g., the _invertible_ ideals are all principal, but
there are non-principal non-invertible ideals like (2, 1+sqrt(-3)).
Z[sqrt(-4)] is a similar example, Z[sqrt(-27)] is another, and I'm
too lazy to discover the fourth ;-). Is this what you had in mind?
[Since Noam took the ring of integers of a number field, that is then
integrally closed by the usual definition. - Greg]
|> The Euclidean
|> domains with both d<0 and d>0 are known, but I don't remember the list
|> or a good reference. It's conjectured that there are infinitely many
|> UFD's with d>0 but there's probably no proof coming soon. --NDE]
In a followup article (which I haven't seen here yet), Bob Silverman
(bobs@mathworks.com) provided the reference to Hardy & Wright and
the list of norm-euclidean quadratic fields, and in article
<199506172018.NAA20568@ucdmath.ucdavis.edu>, dean@ucdmath.ucdavis.edu
(Dean Hickerson) commented on this as follows:
|> This is only partially correct. The general definition of a Euclidean
|> ring is that there is some function n from the nonzero elements of the
|> ring to the nonnegative integers for which the division algorithm works.
|> [...] It
|> happens to be true that for every algebraic number field that's been
|> proved to be Euclidean we can take n to be the absolute value of the
|> norm; i.e. n(a) is the absolute value of the products of the conjugates
|> of a.
This is no longer true; David A Clark recently showed that the ring
of integers of Q(sqrt(69)) , which fails to be norm-euclidean, is
euclidean with respect to a multiplicative function which agrees with
the absolute norm on all primes except that of norm 23 where it needs to
take a larger value (anything > 25 will do), and Clark and Franz Lemmer-
meyer proceeded to find some more examples like this in higher degrees.
See "A quadratic field which is euclidean but not norm-euclidean" by
D A Clark, Manuscripta Math. 83 (1994), 327--330. See also the note
by Yours Truly, same volume, 443--446. The proofs use (slight variants
of) time-honored number-geometric methods.
[...]
|> In particular, the ring Z[sqrt(14)] is
|> probably Euclidean; in "L'anneau Z[sqrt(14)] et l'algorithme Euclidien",
|> by Edmondo Bedocchi, Manuscripta Math. 53 (1985), 199-216, a function n
|> is given with respect to which the ring is conjectured to be Euclidean.
This ring, however, has so far resisted a determined attack along the
same lines...
Gerhard Niklasch
--
+------------------------------------+----------------------------------------+
| Gerhard Niklasch | All opinions are mine --- I even doubt |
| | whether this Institute HAS opinions:-] |
+------------------------------------+----------------------------------------+
==============================================================================
From: elkies@ramanujan.math.harvard.edu (Noam Elkies)
Newsgroups: sci.math.research
Subject: Re: Euclidean rings (Re: Quadratic Fields)
Date: 20 Jun 1995 00:23:54 GMT
In article <1995Jun19.103103@mathematik.tu-muenchen.de>
nikl@mathematik.tu-muenchen.de writes:
>[...]
>In Z[sqrt(-3)] , e.g., the _invertible_ ideals are all principal, but
>there are non-principal non-invertible ideals like (2, 1+sqrt(-3)).
>Z[sqrt(-4)] is a similar example, Z[sqrt(-27)] is another, and I'm
>too lazy to discover the fourth ;-). Is this what you had in mind?
Yes it is, except that for Z[sqrt(-27)] substitute Z[(1+sqrt(-27))/2].
The fourth example is Z[sqrt(-7)], which is of index 2 in its
integral closure Z[(1+sqrt(-7))/2].
--Noam D. Elkies (elkies@math.harvard.edu)
Dept. of Mathematics, Harvard University
==============================================================================
Date: Mon, 19 Jun 95 13:45:23 CDT
From: rusin (Dave Rusin)
To: katz@galois.njit.edu
Subject: Re: Quadratic Fields
Status: RO
In article <1995Jun16.190331.22984@njitgw.njit.edu> you write:
>Consider the quadratic number fields, a + b * sqrt (d), where
>a, b are rational and d is a square free integer, positive or
>negative. For which d is the field a
>i) Euclidean domain
>ii) Principal ideal domain
>iii) unique factorization domain ?
>
>[The *field* is, of course, a UFD, but presumably you mean the ring
>of integers... Anyway, (ii) <-> (iii). For d<0 it's known that there
>are only 9 fields and 13 rings (four of which are proper subrings of
>the rings of integers) which have unique factorization. The Euclidean
>domains with both d<0 and d>0 are known, but I don't remember the list
>or a good reference. It's conjectured that there are infinitely many
>UFD's with d>0 but there's probably no proof coming soon. --NDE]
I suppose it's possible that you did _not_ mean the ring of integers. The
distinction occurs precisely when d = 1 mod 4. In that case,
the ring of integers of the field Q[sqrt(d)] is the ring of all integral
linear combinations of 1 and (-1 + sqrt(d) ) / 2. You do need to be
careful when reading the literature if for example you really want to
know about Z[sqrt(-3)], since this is only a subring of what most
people are studying (in this case, Z[1^(1/3)].) The reason the larger ring
is preferred is that it is integrally closed (in its quotient field) -- in
this example, for instance, the roots of x^2+x+1=0 are integral over
Z[sqrt(-3)], but lie in the quotient field of this ring.
Usually it's true that unique factorization (and thus also
being a PID) requires that the ring be integrally closed. In the
example above, for example, 2.2=(1+sqrt(-3))(1-sqrt(-3)) defies unique
factorization. Likewise it is clear that d must be square-free.
Here are two of the lists Elkies implied. The data come from Borevich
and Shafarevich (see p. 167, 422, and 425).
When d < 0, any Euclidean algorithm would give a UFD, and there
are only nine such fields, those with -d =
1 2 3 7 11 19 43 67 163
I don't know what the other four rings he mentioned are.
I also don't know just which of these UFDs really are Euclidean --
when d= -1 , -2, or -3, the rings are Euclidean, but not when d=-19
(see Motzkin, "The Euclidean Algorithm", Bull AMS 55 (1949) 1142-6.)
Among positive integers d, the ring of integers of Z[sqrt(d)] is a
Euclidean domain relative to the usual norm (size of a+b sqrt(d) is
|a^2-d b^2|) iff d is one of the following
2 3 5 6 7 11 13 17 19 21 29 33 37 41 57 73
I guess it's true that there could be other norms which suffice to
make other rings Euclidean domains, but that seems unlikely.
There are other rings with d>0 which are UFDs but not Euclidean
(relative to the standard norm anyway), probably infinitely many,
I suppose, starting with d=
14 22 23 31 38 43 46 47 53 59 61 62 67 69 71
and all the UFDs with d>73. I don't know in this case what happens
for non integrally-closed rings.
If you get more detailed information I'd appreciate a copy for my files.
dave
==============================================================================
Date: Wed, 5 Jul 1995 17:10:47 -0400 (EDT)
From: robert katz
To: rusin@math.niu.edu
Subject: Re: Quadratic Integers
Gentlemen:
A while back I posed a question concerning quadratic number fields.
Specifically, for which `d' is the domain of integers --I originally
forgot to specify integers-- in the quadratic field a + b * sqrt (d)
i) euclidean
ii) principal
iii) unique factorization
Thank you all for your help. This seems to be the summary:
If you define a size function to be
abs (a^2 - d * b^2) then the euclidean fields are precisely
those with d = -1 -2 -3 -7 -11 2 3 5 6 7 11 13 17 19 21 29 33 37 41 57 73.
For a different size function, there may be other euclidean domains.
The ufd also include the values of d = -19 -43 -67 -163, and it is
unknown whether there are an infinite number of d > 0 for which the
domain is ufd.
Finally, in a quadratic number field, every ufd is a principal ideal
domain.
Thanks again,
Robert Katz
==============================================================================
From: Paul Pollack
Subject: re http://www.math.niu.edu/~rusin/known-math/95/euclidean.dom
Date: Sun, 07 Mar 1999 13:30:45 -0500
Newsgroups: [missing]
To: rusin@math.niu.edu
I was glancing through your wonderful archives, particularly the page
discussing rings of integers of quadratic fields and when they are
Euclidean domains
and noticed the following:
> When d < 0, any Euclidean algorithm would give a UFD, and there
> are only nine such fields, those with -d =
> 1 2 3 7 11 19 43 67 163
> I don't know what the other four rings he mentioned are.
> I also don't know just which of these UFDs really are Euclidean --
> when d= -1 , -2, or -3, the rings are Euclidean, but not when d=-19
> (see Motzkin, "The Euclidean Algorithm", Bull AMS 55 (1949) 1142-6.)
As mentioned on your page, when d<0 the ring in question is norm
Euclidean for d=-1,-2,-3,-7,-11 and no others. It seems relevant to note
that Dubois & Steger (1958) proved that in the imaginary quadratic field
case, the ring is Euclidean w.r.t. some function <==> it is
norm-Euclidean. Thus for d<0, we have a complete list.
Their proof goes like this: Let R(d) be the ring of integers of
Q(sqrt(-d)), and suppose R(d) is Euclidean with respect to some "norm"
f. Let alpha be a nonzero nonunit which minimizes f, and look at
R(d)/ which has N(alpha) elements. But since R(d) is Euclidean
with respect to f, by D.A. and our choice of alpha, R(d)/ can
have as elements only the cosets of 0 and units. Since we can assume d
!= 1,2,3,7 or 11, the only units are 1 and -1 ==> R(d)/ has at
most 3 elements ==> N(alpha) <= 3. But it is easy to see that for d !=
1,2,3,7,11, there are no elements of norm 2 or 3 in R(d) ==> N(alpha)=1
==> alpha is a unit, contradicting our choice of alpha.
Sincerely,
Paul Pollack
==============================================================================
From: Paul Pollack
Subject: Re: re http://www.math.niu.edu/~rusin/known-math/95/euclidean.dom
Date: Mon, 08 Mar 1999 19:16:46 -0500
Newsgroups: [missing]
To: Dave Rusin
Dave Rusin wrote:
> Thanks for your kind remarks, and your suggestions for improvement.
> Unless you have some specific objection, I'd like to append your remarks
> to the file.
I would be honored. I didn't see the result in question mentioned
already, and it seemed somewhat interesting, so I just decided to chime
in. I hope it's not already somewhere else in your extensive archives!
[deletia --djr]
Sincerely,
Paul Pollack
==============================================================================
From: Franz Lemmermeyer
Subject: Re: Unique Factorization Domains
Date: Sun, 24 Oct 1999 18:11:01 +0200
Newsgroups: sci.math.research
Keywords: Which quadratic extensions of Z are UFDs?
Daz wrote:
>
> I am fascinated by the theorem stating that among the rings Z[sqrt(-m)]
> for m a positive integer, the only ones with unique factorization are
> those for m = 1,2,3,7,11,19,43,67, or 163.
>
> QUESTION 1: Is there some deep reason that the successive differences of these
> numbers divide each other (with all successive quotients being 1, 2, 3, or 4)?
>
> That is, 1 | 1 | 4 | 4 | 8 | 24 | 24 | 96.
1. It is not the rings Z[sqrt(-m)] that are UFD's but the rings
of integers in Q(sqrt(-m)): these are different from Z[sqrt(-m)] for
m = 3 mod 4.
2. Yes there is a "deep" reason for this kind of behaviour: the integers
19,43,67, 163 are quadratic nonresidues modulo small primes (and must
be
if Q(sqrt(-m) is to have class number 1). This forces these integers to
lie in certain residue classes modulo the product of small primes.
> QUESTION 2: What is known about which of the rings Z[sqrt(m)], for m
> a positive integer, have unique factorization?
Gauss conjectured that there are infinitely many. See Hardy & Wright,
Ireland & Rosen, or one of H. Cohn's books on (algebraic) number theory.
franz
==============================================================================
From: "Daniel Grace"
Subject: Conditions on d for {a+b*sqrt(d)} a UFD?
Date: Wed, 17 Nov 1999 09:14:30 -0000
Newsgroups: sci.math
What are conditions must d satisfy for
F={a+b*sqrt(d):a,b are integers} to be unique factorization
domain?
Obviously F contains complex numbers iff d is negative.
I have seen a complete list of negative values for d in books
and on the following web page
http://www.ams.org/new-in-math/cover/factorization.html
The theorem that proves this list is complete is "deep".
Presumably that means there are lots of cases and
a long proof?
For what positive d's if any is F a UDF?
What are the neccessary conditions for positive d?
Thanks,
----------------------------------------------------------
Daniel
e-mail: dan.@landemann.freeserve.co.uk
==============================================================================
From: Robin Chapman
Subject: Re: Conditions on d for {a+b*sqrt(d)} a UFD?
Date: Wed, 17 Nov 1999 21:45:33 GMT
Newsgroups: sci.math
In article <80tsre$bdr$1@news6.svr.pol.co.uk>,
"Daniel Grace" wrote:
> What are conditions must d satisfy for
> F={a+b*sqrt(d):a,b are integers} to be unique factorization
> domain?
> For what positive d's if any is F a UDF?
> What are the neccessary conditions for positive d?
A lot of examples are known, but it's not known whether
there are finitely or infinitely many instances.
--
Robin Chapman
http://www.maths.ex.ac.uk/~rjc/rjc.html
"`Well, I'd already done a PhD in X-Files Theory at UCLA, ...'"
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