The Dirichlet Class Number Formula for Imaginary Quadratic
Fields
HuiBin Liu*
Department of Commercial Design
Chienkuo Technology University
No. 1, Chieh Shou N. Rd., Changhua City 500
Taiwan
Abstract
We prove a formula for the average value of L-functions associated to quadratic
function fields (with constant field of odd characteristic ramified at one finite place and 1,
which we will call "imaginary prime quadratic function field." In particular, we obtain a
formula for the average class number over the associated rings of integers using the
L-function average at s = 1.
Keywords : Golden section, real quadratic fields, polynomial complexity.
1.
Introduction
Fermat stated that every rational prime p congruent to 1 modulo 4 is the
sum of two squares of natural numbers. Fermat’s theorem has received many
proofs [22, p. 66], although early ones did not offer a method of deterministic
polynomial complexity for computing the representation p = s2 + r2, i.e. one
using a number of sums and products proportional to some small power of the
logarithm of p. However, a reduction algorithm of binary quadratic forms,
which can be traced back to Gauss [6], produces r and s with deterministic
polynomial complexity once a square root of −1 modulo p is known [13]. The
question was then settled by Schoof, who discovered an algorithm for
evaluating the number of points on elliptic curves over finite fields, which also
furnishes a square root of small numbers, in particular −1, modulo p with
*E-mail:
eghm12@tin.it
—————————————————–
deterministic polynomial complexity [18]. Fermat also stated, and Lagrange
proved, that every natural integer is the sum of at most four squares of natural
numbers [17, p. 7]. Some two centuries later, Siegel proved that the only real
algebraic number fields in which exist a four-square-sum representation of the
totally positive integers are the rational field Q and the quadratic field Q(𝜔)
with golden section unit ω =
1+√5
2
, [20]. A direct consequence of Siegel’s
theorem is that any prime integer p ∈ Q(𝜔) has an associated prime which is
the sum of two squares in the same field, if its field norm is a rational prime not
congruent to 11 or 19 modulo 20.
The paper shows that the same tools used in Q can be employed to give a
proof of the two-square-sum property for primes 𝜋 in Q(𝜔), and provides a
way of computing their representation 𝜋 = a2 + b2 with deterministic
polynomial complexity.
2.
Preliminaries
The real quadratic field F = Q(𝜔) is a principal ideal domain whose
integers are the elements of Z (𝜔), [5]. The conjugate of an element z = a + b𝜔
is 𝑧̅ = a + b𝜔
̅ = (a + b) − b𝜔, and the field norm is NF(z) = z𝑧̅ = a2 + ab −
b2. The elements of Z(𝜔) with norm ±1 are called units, and form a group
which is the direct product of a torsion group T2 = {1, −1} of order 2, and an
infinite group U of rank 1, which is generated by 𝜔. Two integers α, β ∈
Z(𝜔) are said to be associated if their ratio 𝛼/𝛽 is a unit. A positive integer
number 𝛼 = a + b 𝜔 ∈ Z(𝜔) is said to be totally positive if its conjugate a is
also positive [15, p. 378]. Given 𝛼, one of the four associates 𝛼, − 𝛼, 𝛼 𝜔,
and − 𝛼 𝜔 is certainly totally positive; moreover, let U2 be the subgroup of U
generated by 𝜔2, every totally positive element 𝛼 ∈ Z(𝜔) uniquely identifies
a class 𝛼U2 = {𝛼u : u ∈ U2} of totally positive associates.
The set of primes in Q(ω) consists of: (i) a prime π5 = 2 + ω of field
norm 5, and its associates; (ii) all rational primes q ≡ ±2 mod 5, and their
associates; and (iii) all pairs of prime factors 𝜋q = a + b ω and 𝜋̅q = a + b −
b 𝜔 of every rational prime q ≡ ±1 mod 5, and their respective associates [5,
p. 25], in particular πq may be chosen totally positive and such that q = 𝜋q𝜋̅q =
a2 + ab − b2. This explicit representation is obtained from the process that
reduces the quadratic form
pX2 + s5 XY +
𝑠52 −5
4𝑝
Y2
27
of discriminant 5 to the canonical form X2 + XY − Y2, where s5 is the odd
root of x2 = 5 mod p.
This reduction algorithm, outlined by Gauss and cleverly reported in
Buell [3], was described in a masterly fashion in Mathews’ book [13], and thus
it will be referred to as Mathews’ reduction. It can be applied to any quadratic
form of a non-perfect square discriminant. The square root s5 may be computed
with deterministic polynomial complexity by distinguishing whether p is
congruent 3 or 1 modulo 4. In the first case, s 5 is one of the two square roots
p+1
±5 4 mod p, and the computation clearly has deterministic polynomial
complexity. In the second case Schoof’s algorithm [18, 21] is applied to
compute both √−1 and √−5
modulo p, thus s5 is one of the two values
±√−1√−5 mod p. Schoof’s algorithm evaluates a square root modulo p in a
number of arithmetical operations of order O(logt p) with t ≤ 10, that is with
deterministic polynomial complexity [18]. A description of Schoof’s algorithm
can be found in the original paper [18] or in Washington’s book [21], and is
based on properties of elliptic curves that are briefly summarized.
........
3.
Representation of primes in Z(𝝎)
4.
Computational remarks
........
The representations of primes 𝜋 ∈ Q(ω) as the sums of two squares,
established by Theorems 2 and 3, are computed with deterministic poly-nomial
complexity using procedures that will be described separately for the different
cases.
4.1
Primes p ≡ 13, 17 mod 20
The representation p = a2 + b2 does not need any further comment, since
p = 1 mod 4 is the sum of two integer squares by Fermat’s theorem, and its
computation was shown in Section 2.
4.2
Primes p ≡ 1, 9 mod 20
u0 mod 5
x0 mod 5
y0 mod 5
0
0
≠0
0
≠0
L1 = 0 mod 5 ?
x0 = 2y0
k
𝐿′1 mod 5
0
2 y0 ≠ 0
3
2 y0 ≠ 0
≠0
0
≠0
≠0
𝐿′1
≠0
u0 = −2y0
1
2 y0 ≠ 0
u0= x0 − 2y0
2
x0 ≠ 0
= −( x0F2k-1 + y0 F2k) + u0 +2(x0F2k + y0F2k+1)
......
5.
Conclusions
The representation of primes as sums of two squares in Z(𝜔) allows us to
formulate a theorem concerning the representation of any integer in Z(𝜔):
Theorem 4. In the field Q(𝜔), any totally positive integer
.....
The computation of the two-square representation of any element in Z(𝜔)
is equivalent to factoring, as in Z, therefore at the present time only the
two-square representations of primes can be computed with deterministic
polynomial complexity.
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Received October, 2005