Re: norm of fundamental unit

[Bill Allombert <Bill.Allombert@math.u-bordeaux1.fr>]

On Tue, Mar 09, 2010 at 02:43:47PM -0500, Max Alekseyev wrote:
> I apologize if my question is not directly related to PARI.
> It is about quadratic fields Q(sqrt(d)) where d is a positive
> non-square integer. I wonder how to classify such d that there exists
> an element of norm -1, while the norm of fundamental unit (computed by
> PARI) is 1?
> 
> An example is given by d=34:
> 
> ? bnfinit(x^2-34).fu[1]
> %1 = Mod(6*x + 35, x^2 - 34)
> ? norm(%)
> %2 = 1
> ? norm(Mod(1/5*x-3/5,x^2 - 34))
> %3 = -1

Well, Mod(1/5*x-3/5,x^2 - 34) is not an algebraic integer.  Finding such
element of norm -1 is equivalent to solving the equation x^2-d*y^2=-4*z^2 which
is equivalent to x^2+4*z^2=d*y^2 which has solutions if and only if d is the
sum of two squares (i.e has no prime divisors congruent to 3 mod 4).  

On the other hand, the norm of the fundamental unit is an arithmetic
invariant with a poorly understood behaviour and there is an extensive
litterature on the subject. It is linked to the 2-part of the class group.
Due to the above condition, it is often 1, except when the field discriminant
is prime in which case it is always -1.

Cheers,
Bill.