Integer Rings

Date: 05/18/2005 at 02:35:12
From: Francois
Subject: "integer" ring in a field

Given a (commutative, characteristic zero) field F, is (are) there any 
ring(s) such that F is the fraction field of R? If any, how many, and 
how are they related?

The characteristic zero case is the one I'm faced with, but 
characteristic p will soon creep in, and is obviously interesting in 
its own right.

This is sort of "backward" construction of the fraction field of a 
ring.  While I can certainly consider the ring I of integers (in F) 
over Z, F is usually not "simply" the fraction field of I.  I think
this is a classical problem, but I couldn't find anything similar in
Lang, Hasse, Shafarevich, and many others.  Just a pointer to some
reference would be greatly appreciated.

In characteristic zero, F contains (as its prime field) the field Q of 
rationals, and thus is an extension of the fraction field of Z.  It 
must be of the form Q[x] for some irrational x (the "generator") but 
I cannot "go back" to something analogous to Gauss's integers or the 
like.

Date: 05/18/2005 at 16:30:57
From: Doctor Vogler
Subject: Re: 

Hi Francois,

Thanks for writing to Dr. Math.  That's a good question.  But let's 
take things a little more slowly, and think about exactly what you mean.

Consider the field of rational numbers Q.  You already know that the
ring of integers is Z, right?  But wait!  What if we let R be the ring
of even integers?  Now what is the field of fractions?  Well, if a/b
is a rational number in Q, then it is the same as (2a)/(2b), which
means that every rational number is in the field of fractions of R.

So there are two possible rings of integers.  Are there more? 
Absolutely!  For example, we could choose any integer n and then let R
be the ring of rational numbers whose denominators are powers of n. 
Its field of fractions is also Q.  Or we could let R be the "local
ring" of a prime number p, which is the ring of rational numbers whose
denominators are NOT divisible by p (that is, the denominators are
relatively prime to p).  So there are infinitely many possibilities
for the ring of integers, and why should we choose the ring Z over
another ring, such as nZ for some integer n, or a local ring?  And
there is one very important ring whose field of fractions is Q, and
that ring is Q.  Why shouldn't we choose Q?

Now let's back up and take a different approach to things.  This is
the approach taken in number theory, and has to do with the topic of
number fields.  A number field is a finite (i.e. algebraic) extension
of the field Q of rational numbers.  For example, Q(sqrt(2)) is a
number field.  Notice that, since they are finite extensions of Q,
every element of a number field is the root of some polynomial with
rational coefficients.  By multiplying by a common denominator, every
element of a number field is the root of some polynomial with integer
coefficients.  We call an algebraic number an "algebraic integer" if
it is the root of some MONIC polynomial (leading coefficient 1) with
INTEGER coefficients.  For example,

  sqrt(2)

and

  6 - 7^(1/3)

are algebraic integers.  (Can you find their polynomials?)  But

  sqrt(2)/3

is not.  Furthermore, the rational numbers that are algebraic integers
are precisely the normal, regular integers that we are all so familiar
with.

The ring of integers R of a number field F is the set of algebraic
integers contained in F.  This ring R has some very nice properties
that make it useful to study.  In particular, it is a ring (though
it's not completely trivial to show that the sum or product of two
algebraic integers is again algebraic), and its field of fractions is
the number field F.  So it might be considered "the" ring of integers
of the field F.

Now let's think about this and try to extend it to more general 
fields.  The whole idea was based on the minimal polynomial over the
base field of rational numbers, and a pre-determined concept of
"integer" in that field.  That would imply that we should start by
defining integer in a small field and then look at extensions.

But if we look at the finite field with p elements, sometimes denoted
GF(p), then there is really no reason to expect one of those p 
elements to be an integer more than any other.  In fact, if any 
nonzero element of the field is considered an integer, and you want
"integers" to be a ring, then twice that number must be an integer,
and three times that number, and all of GF(p).  Of course, GF(p) is a
ring, and its field of fractions is GF(p).  So for some fields, there
is no better choice than the whole field.

But there is another field that is strangely similar to the rational
field Q, and that field is GF(p)(x), a transcendental extension of
GF(p) of transcendental degree 1.  That is, the field of rational
functions or rational polynomials in x with coefficients in the finite
field GF(p).  In this field, the natural idea to use for "integers" is
those rational functions whose denominators are constant; that is, the
"integers" are the polynomials in x.  Then you define integers in
finite extensions of GF(p)(x) just like for number fields, namely
roots of monic polynomials with "integer" (polynomial) coefficients.

If you have any questions about this or need more help, please write
back and show me what you have been able to do, and I will try to
offer further suggestions.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 05/26/2005 at 15:38:12
From: Francois
Subject: "integer" ring in a field

Dear Dr. Vogler,

I sincerely thank you for your quick and enlightening answer to my 
question.

As often happens to me, my confusion came from the fact that my 
problem was not properly stated.  In fact, I was looking for some
"canonical, smallest" ring R that has a given field F as its fraction
ring.  Of course, neither "canonical" nor "smallest" are correctly
specified, hence my question.

Taking the example of the rationals Q (my starting point, in fact), I 
was very aware that any subring of Z does the job.  But rings of the
form nZ, I do not consider to be essentially different from Z, bacause
of "canonical" isomophism.  Maybe the rings pZ (p a prime) are 
"better", since they are maximal ideals in Z ; of course, having
"smallest" mean "maximal" may bother some people, but I'm ready to
reformulate my question if it does bring more rigor.

I didn't at first think of local rings (my knowledge in number theory 
is rather superficial) but I did find they were also a possibility.

Of course, having the whole field F as an answer is not satisfactory 
(not to say, somewhat dishonest); although, as you mention in the case
of GF(p), there may be no better answer.

Now, for the rest of my reasoning, let me call "zintegers" the 
elements of Z.  Then the operation k.x = x + x + ... + x (k times) 
does make sense whatever the field F.  So, defining algebraic integers 
as the roots of some monic polynomial with zintegers coefficients does
make sense.  And, with Q, I get the classical integers Z; as with
p-adic numbers Q_p, I get p_adic integers Z_p, which was what I was
looking for.

All this sounds right.  But I don't quite understand you (well, I 
think I do, but I'm not sure this is what I'm looking for) when you
consider the field GF(p)(X).  As you mention, the problem is to have a
pre-defined concept of "integer" in a field, and investigate extensions.

What worries me is that it sounds like it amounts to decree who 
deserves the title of "integer", and then investigate who deserves it
"by alliance", i.e., you decree who is noble, and then decree that
anyone with at least a male noble ancestor is also a noble.  I don't
like this.  "Integers" should have a "natural" definition (maybe in 
the sense of categories, but this would be clear overkill here); 
considering "vaster" rings may be more constructive mathematically
but make me feel morally uncomfortable.

I would appreciate some more enlightment, or references.  I thank you
again for shedding some (very helpful) light on my problem.

Yours sincerely,

-- Francois

Date: 05/26/2005 at 19:53:54
From: Doctor Vogler
Subject: Re: 

Hi Francois,

Okay, I will analyze the reformulation of your question.

>I didn't at first think of local rings (my knowledge in number theory
>is rather superficial) but I did found they were also a possibility.
>
>Of course, having the whole field F as an answer is not satisfactory 
>(not to say, somewhat dishonest); although, as you mention in the
>case of GF(p), there may be no better answer.
>
>Now, for the rest of my reasoning, let me call "zintegers" the 
>elements of Z.  Then the operation k.x = x + x + ... + x (k times)
>does make sense whatever the field F.  So, defining algebraic
>integers as the roots of some monic polynomial with zintegers 
>coefficients does make sense.  And, with Q, I get the classical 
>integers Z; as with p-adic numbers Q_p, I get p_adic integers Z_p, 
>which was what I was looking for.

Um, actually you don't.  I believe that you *do* get nothing but 
p-adic integers, although this would require some proving, but you
certainly don't get *all* of the p-adic integers.  You see, there are
uncountably many p-adic integers, but there are only countably many
roots of polynomials with "zinteger" coefficients.

Furthermore, the field of fractions of the ring of "algebraic" p-adic
integers, as you defined it, would not be all of Q_p.  Again, it would
only be a countable set, sort of like the set of algebraic numbers
does not include all of the complex numbers.

>All this sounds right.  But I don't quite understand you (well, I 
>think I do, but I'm not sure this is what I'm looking for) when you
>consider the field GF(p)(X).  As you mention, the problem is to have 
>a pre-defined concept of "integer" in a field, and investigate
>extensions.
>
>What worries me, is that it sounds like it amounts to decree who 
>deserves the title of "integer", and then investigate who deserves it
>"by alliance", i.e., you decree who is noble, and then decree that
>anyone with at least a male noble ancestor is also a noble.  I don't
>like this.  "Integers" should have a "natural" definition (maybe in
>the sense of categories, but this would be clear overkill here); 
>considering "vaster" rings may be more constructive mathematically
>but makes me some feel morally uncomfortable.

Yes, well, "should" is also up to debate, as you have to declare who
can decide what should be and what shouldn't.

As it happens, integers were defined before rationals, and rationals
were defined as ratios of integers, so it shouldn't be too hard to
understand that fields come out of rings rather than vice-versa. 
Sometimes, there is a way to reverse a kind of inheritance like this,
but sometimes there isn't.

For example, the integers Z map into the integers mod n Z/nZ, in a
very natural way.  It is easy to turn an integer into an integer mod
n.  But if you have an integer mod n, can you determine what integer
it started out as?  The answer is no; you can't.  Something similar is
happening here.  So you can choose "special" integers, like the ones
from 0 to n-1, and say that the integers mod n come from those, and
thus they are the "noble" integers, and so our map from Z/nZ into Z
should go there.  But someone else might say that -1 should also be a
"noble" integer and suggest something on the order of -n/2 to n/2 to
be a better range to us.

You, too, are looking for a map from the class of fields to the class
of rings which has certain properties, and there are just too many
ways to define such a map, which means that making it well-defined
requires choosing your "noble" rings.

- Doctor Vogler, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 05/28/2005 at 03:31:27
From: Francois
Subject: "integer" ring in a field

Dear Dr. Vogler,

I thank you again for you quick and useful answer.  I can now re-read 
my books in Algebra and Number Theory with a different view and
understanding.

I'm sorry I've been a bit hasty about p-adic numbers, but that was 
only meant to be an example of what I was looking for.  Of course, you
are perfectly right in your remarks.

Now, my problem is clearly what we call in applied mathematics (my 
actual job, in fact) "underspecified".  There is no "better" solution
as long as you do not define precisely what "better" means.  While I
certainly understand that looking for a "natural" set of "integers" in
a given field is unrealistic, I however hoped that some (not too 
severe) restrictions on "natural" would yield a (severe) limitation on
the possible solutions.

Your examples and explanations show clearly that this is not generally 
the case.  In my particular problem, divisibility is anessential 
property, so the natural setting is that of p-adic numbers.  It would 
(probably?) not be the same from the point of view of, say, algebraic 
geometry or topology. 

In applied mathematics, it is not the mathematician who dictates what
"should" be natural: it is the phenomena under study.  Thus, I have to
keep modest and not superimpose some "theoretical" point of view just
to make things look "elegant"--after all, I'm not Bourbaki!

Thank you once more.  Now I have some books to re-consider...

Yours sincerely,

-- Francois