Bill Allombert on Tue, 09 Mar 2010 21:40:10 +0100 |
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Re: norm of fundamental unit |
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.