Rational equivalence of unimodular circulants

Abstract

Let P be the n X n matrix satisfying P I2= . . ,= P" _ I , I/ = P " , 1= 1 a nd all other entries O. The elements of the group rin g Z [P] are caJl ed integral circ ulants. Let G be the group in Z [P] consisting of the pos itive de finite symme tric unimodula r ele ments. M. Ne wma n [2, p. 198] as ks whi ch me mbers of G are ra tionall y equivalent to [ 1/. Th e a nswer to thi s question is in fact a n easy co nseque nce of th e Hasse norm theore m [1 , p. 186]. , TH EO REM : A LL members of G a re rat iona LLy equivalent to In. PROOF: Let A EG. As noted in [2 , p. 198] , we must show that an y eigenvalue A of A is of the form aa for some a in QI/' the nth cyclotomic fi e ld . Le t K be the real s ubfield of index 2 in Q". We must show A is a norm from Q" to K. If n = 2p ", p a prime, the n p is fully r amifi ed in Q 1/. I n any othe r case, Q" is unramified over K at all finit e primes. So at most one finite prim e ramifies fro m K to Q". N ow A is totally positive, so A is a norm at all Archimedean localiza tions. And A is a unit ; thu s A is a norm at all finite localizations, except , poss ibly, one. By the product formula , A is a norm ever ywhe re, and hence, by th e Hasse norm theore m, A is a global norm .