Bill Allombert on Sun, 5 Oct 2003 14:37:19 +0200 |
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Re: Cyclotomic Integer with a Given Norm? |
On Sun, Oct 05, 2003 at 03:42:36PM +0800, B.Y. wrote: > Dear Sirs (and Mesdames): > > Can someone tell me if there is a ready and easy way to decide > with PARI whether there exists an integer $a$ in a cyclotomic field > $Q(\zeta_n)$ with a given norm $p$, where $p$ is a prime number? If I am not mistaken, your problem is equivalent to deciding whether 1) the ideal above p are of norm p and 2) the ideals above p are principals. You can do: ? B=bnfinit(polcyclo(n)); ? L=idealprimedec(B,p)[1]; ? M=bnfisprincipal(B,L) p is the norm of an integer of B iff L.f==1 and M[1]==0, the integer being M[2] (expressed on the integral basis). The above work because Q(\zeta_n) is Galois and totally complex (for n>2). An alternative solution that look simpler but may not do what you want: ? N=bnfisnorm(B,p,0) p is a norm of a element N[1] of B iff N[2]==1 under the GRH, but N[1] is not warranted to be integral. Cheers, Bill.