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A quadratic field is a field F which has dimension two (when considered as a vector space over the rational field Q). A quadratic field is imaginary if it cannot be embedded in the real field R
Since we'll only consider imaginary quadratic fields, so below "field" always means "imaginary quadratic field." Some definitions and theorems hold in greater generality, but the Lounge doesn't care
An element r in the field F belongs to the ring of integers O(F) if r is a root of some monic polynomial with ordinary integer coefficients. "The ring of integers O(F)" really is a ring. Moreover, if r belongs to the ring O(F), then r actually satisfies a monic quadratic polynomial with ordinary integer coefficients: that is, we can find ordinary v and w (both ordinary integers) such that r^2 + u*r + v = 0
A fractional ideal J is a nontrivial and nonempty finitely-generated O(F)-submodule of F with the following additional property: there exists some nonzero r in O(F) such that rJ is an ideal in the ring O(F). If f is a nonzero element of F, then fO(F) is called the principal fractional ideal generated by f. Here rJ, for example, denotes the collection of all products r*j with j in J; similarly, fO(F) denotes the set of all products f*s with s in O(F). Notice that every "principal fractional ideal" is a fractional ideal
If I and J are fractional ideals, the "product of I and J" (denoted by IJ) the collection of all finite sums of elements i*j with i in I and j in J. This product operation makes the set of fractional ideals into an abelian group, in which the principal fractional ideals appear as a subgroup
The quotient of the group of fractional ideals by the principal fractional ideals is called the ideal class group. The ideal class group is finite
The size of the ideal class group is called the class number C(F). O(F) is a principal ideal domain if and only if C(F) = 1. Being geeks, we are therefore very interested to compute the class number. And, due to the nineteenth century heroism of Dirichlet, we have a formula for the class number!
Let n be a natural number. A Dirichlet character X is a map from the multiplicative semigroup Z/nZ (ordinary integers mod n) to the multiplicative semigroup C (complex numbers) of the following form: restricted to the group of units of Z/nZ, X is a group homomorphism to the nonzero complex numbers, and X is zero everywhere else. In other words, if p and q are relatively prime to n, the X(p*q mod n) = X(p mod n)*X(q mod n) with all terms nonzero; while if (n,p) > 1, we have X(p) = 0
Let X be a Dirichlet character. The Dirichlet L-function L(s,X) is defined by the infinite series
L(s,X) = X(1)/1^s + X(2)/2^s + X(3)/3^s + ...
The Legendre symbol is defined as follows: (a/p) = 1 if the equation x^2 = a mod p has a solution, and otherwise (a/p) = -1
Our field F, being imaginary quadratic, is obtained from Q by adjoining the square root of a negative squarefree integer -m. Define N = m if -m = 1 mod 4 and N = 4*m otherwise. We now produce and use a particular Dirichlet character, continuing to write X because this is the only Dirichlet character we will consider:
X(q mod N) = t(q)*(product of all Legendre symbols (q/p) where p is an odd prime dividing m)
where t(a) is defined as follows:
if -m = 1 mod 4 then t(a) = 1
if -m = 3 mod 4 and a = 1 mod 4 then t(a) = 1
if -m = 3 mod 4 and a = 3 mod 4 then t(a) = -1
if -m is even and (a = 1 mod 8 or a = 1 + m mod 8) then t(a) = 1 else t(a) = -1
Class Number Formula The class number is w(F)*sqrt(N)*L(1,X)/(2*pi) where w(F) is the number of roots of unity in the field F
Taking F = Q(i), we have O(F) = gaussian integers. Here, -m = -1 = 3 mod 4. Clearly, N = w(F) = 4 and sqrt(N) = 2. There are no odd primes dividing m, so X(q) = t(q) using the usual convention that an empty product represents 1
and of course in this case t(q) = 1 when q = 1 mod 4, t(q) = -1 when q = 3 mod 4
The formula tells us that the class number is
4*2*L(1,X)/(2*pi) = (4/pi)*L(1,X)
= (4/pi)*(X(1)/1 + X(2)/2 + X(3)/3 + ... )
= (4/pi)*(X(1)/1 + X(3)/3 + X(5)/5 + X(7)/7 + ... )
= (4/pi)*(1 - 1/3 + 1/5 - 1/7 + ... )
even index terms dropping out since the index is not invertible mod 4
Left side is an integer, so it would be enough to compute the right side to an accuracy of better than 0.5; but in this case, we can actually just call on Leibniz. So the class number is one, and the Gaussian integers is a principal ideal domain. QED
Discussion based on: Number Theory I: Fermat's Dream, Kazuya Kato et al, Translations of Mathematical Monographs #186, AMS, 2000
The original Dirichlet-Dedekind (translated by J Stillwell) is: Lectures on Number theory, PGL Dirichlet, History of Mathematics #16, AMS, 1999
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