From: Robin Chapman Subject: Re: Quadratic units problem Date: Fri, 03 Dec 1999 10:04:30 GMT Newsgroups: sci.math Keywords: identification of units in a real quadratic number field In article <3aA14.31\$6c6.6744@den-news1.rmi.net>, Kurt Foster wrote: > I'm sure I've seen this - somewhere - but I can't remember where, and > can't seem to club it into submission easily. It's a matter of showing > that two different expressions, each giving a unit in the same real > quadratic field, both give the SAME unit. > > I know that, if m is odd and squarefree, and m == 3 (mod 4) then the > discriminant D of R = Z(sqrt(m)) is D = 4m. Let X() be the primitive > quadratic character (mod D). Then I know that, if u > 1 is the > fundamental unit of R, and > > U = [PRODUCT, sin(pi*b/D)]/[PRODUCT, sin(pi*a/D)] > > the products being over integers 1 =< a, b < D/2, X(a) = +1, X(b) = -1, > then > > U = u^h > > where h is the class number of R. > > But I *also* know that > > U' = PRODUCT, (1 - 2*pi*i*b/D); 1 =< b < D, X(b) = -1 You probably want 1 - exp(2 pi i b/D) here? > is a unit of R; this is VERY easy to prove. But it doesn't seem so easy > to tell when U' = U. It does when m = 3 and m = 7. Does U' = U always? > I wind up having to decide whether an expression like > > PRODUCT, 2*sin(pi*2*a/D), 1 =< a < D/2, X(a) = +1 > > is equal to 1 (or perhaps -1, I was too lazy to bother with that detail). All the terms in the product are positive, so it can't equal -1 :-) > Anyone have a reference that treats this? Thanks. I don't know about a reference, but let P denote the product. Then sin(2 pi a/D) = sin(2 pi (2m - a)/D) and X(2m - a) = X(a). Thus P^2 is the product of all 2 sin(2pi a/D) over a between 0 and D/2. coprime to D. If we extend the range to between 0 and D we get P^4. Now for any N it's not too hard to compute the product of sin(2 pi j/N) over all 0 < j < N. Then one can use inclusion/exclusion to get at P^4. -- Robin Chapman http://www.maths.ex.ac.uk/~rjc/rjc.html "`Well, I'd already done a PhD in X-Files Theory at UCLA, ...'" Greg Egan, _Teranesia_ Sent via Deja.com http://www.deja.com/ Before you buy.