From: Robin Chapman
Newsgroups: sci.math
Subject: Re: class numbers
Date: Mon, 14 Dec 1998 14:20:20 GMT
In article <19981213191600.05405.00002367@ng34.aol.com>,
finalfntsy@aol.com (FinalFntsy) wrote:
> What is a class number? All I know is that they are sometimes represented as
> h(n), and I think that h(-163)=1 and is the smallest number for which h(n)=1.
One way of defining class numbers is by ideal theory in rings of
algebraic integers. However this type of class number can be defined
by counting equivalence classes of quadratic forms.
A (binary integral) quadratic form is an expression ax^2 + bxy + cy^2
where a, b and c are integers, and its discriminant is b^2 - 4ac. For
instance 3x^2 - xy + 5y^2 is a quadratic form of discriminant -59.
If Q = ax^2 + bxy + cy^2 is a quadratic form, and p, q, r, s are integers
with ps - qr = 1 then Q' = a(px + qy)^2 + b(px + qy)(rx + sy) + c(rx + sy)^2
is also a quadratic form with the same discriminant as Q. We say that
Q and Q' are equivalent forms. The class number h(n) of a discriminant n
is the number of equivalence classes of quadratic forms of discriminant n
if n > 0 and the number of equivalence classes of positive definite
forms of discriminant n if n < 0 [positive definite effectively means that
a > 0]. (Always h(n) is finite.)
For n < 0 the number h(n) can be calculated easily. A reduced quadratic form
of negative discriminant is a positive definite form ax^2 + bxy + cy^2 where
(i) -a < b <= a, (ii) a <= c (iii) if a = c then b >= 0. Each ewquivalence
class of positive definite forms contains exactly one reduced form, so that
h(n) for negative n is the number of reduced forms of discriminant n.
For example when n = -59, the reduced forms are x^2 + xy + 15y^2,
3x^3 + xy + 5y^2 and 3x^2 - xy + 5y^2 so h(-59) = 3. If n = -163
the only reduced form is x^2 + xy + 41y^2 so h(-163) = 1.
For more details see Davenport's "The Higher Arithmetic" of Cox's
"Primes of the Form x^2 + ny^2".
Robin Chapman + "They did not have proper
SCHOOL OF MATHEMATICal Sciences - palms at home in Exeter."
University of Exeter, EX4 4QE, UK +
rjc@maths.exeter.ac.uk - Peter Carey,
http://www.maths.ex.ac.uk/~rjc/rjc.html + Oscar and Lucinda, chapter 20
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From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: class numbers
Date: 14 Dec 1998 20:28:39 GMT
FinalFntsy wrote:
>What is a class number? All I know is that they are sometimes represented as
>h(n), and I think that h(-163)=1 and is the smallest number for which h(n)=1.
Well, it's rather complicated -- why do you want to know?
The simplest answer is that it's a positive integer associated to certain
number system; the bigger the number, the further the number system is from
having unique factorization into primes. Thus "h=1 for the integers" is
equivalent to the Fundamental Theorem of Arithmetic.
A reference to "h(n)" for an integer n probably means the value of h
associated to the set of numbers of the form a + b sqrt(n) where
a and b are integers (in the case n = 1 mod 4, it's these numbers together
with the numbers of the form (a + 1/2) + (b + 1/2) sqrt(n). )
dave
==============================================================================
From: rusin@vesuvius.math.niu.edu (Dave Rusin)
Newsgroups: sci.math
Subject: Re: class numbers
Date: 22 Dec 1998 14:39:55 GMT
FinalFntsy wrote:
>What is a class number?
>rusin@vesuvius.math.niu.edu (Dave Rusin) wrote:
>> ...the bigger the number, the further the number system is from
>> having unique factorization into primes.
Gerry Myerson wrote:
>Yes, I remember being told that, but what does that really mean?
>In what sense is a system with class number 17 farther from having
>unique factorization than a system with class number 12?
What do you want, precision? This is USENET, fer cryin' out loud! :-)
Gerry's point is well taken. Here's a response which isn't all that useful
but maintains the spirit of the my original post.
Suppose R is a number field with class number h. Let K be its
quotient field. Then we may select integral ideals A_1, ..., A_h
to represent the classes: every other ideal now has a representation
I = A_i (k) for unique some i <= h and some k in K which is unique up
to units. (Better: we can stay integral. For each I there is an i and
an r _in R_ with A_i I = (r). )
Now, each ideal I also has a unique decomposition into prime ideals
I = P_1 ... P_t. As above, we deduce factorizations A_i P = (r) for
every prime ideal P; we can call the ring elements r "pseudo-prime" or
something. (The designation of which ring elements are pseudo-prime
depends on our choice of class representatives A_i.) Then for every ideal
we deduce there is a unique factorization
(A_{i_1} ... A_{i_t}) I = (r_1) ... (r_t)
into pseudo-prime principal ideals.
If I itself is principal, it follows the product of the A_{i_j} is, too.
We can collect together the minimal such products of the A_i's (roughly:
find a basis for the appropriate kernel of a map of Abelian groups)
so as to find ring elements b_i with the following property: for
every element s of R there is a unique factorization
b_{i_1} ... b_{i_k} s = u r_1 ... r_t
for some pseudo-primes r_i, some unit u, _and_ some collection of the
b_i from among this pre-determined set.
The class number determines the number of A's and hence the number of
fudge factors b, so it is not unreasonable to say a larger class number
represents a larger deviation from unique factorization.
Note: although I claim uniqueness for some of these derived representations,
I'm not really convinced of that myself; still it seems pretty clear that
the larger h's will require more b's in general.
The right way to view the constructions I have given is that the class
group is the tool needed to decide how many primes must be inverted to
embed R in a unique factorization domain R' intermediate between
R and K. But there again it isn't quite the number h which pops out
in general.
dave
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