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Old 2017-09-18, 05:21   #3
carpetpool
 
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"Sam"
Nov 2016

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Thanks, Dr Sardonicus. I found a similar result with K = Q(\zeta_{29}), and the program returned this: (note I made an additional script file for this):

parinorm.txt:

normU(k,n,m) =
{
bnf = bnfinit(polcyclo(k));
v=[];
w=[];
j=0;
l=0;
forprime(p=n,m,
if(p%k==1,j++;
if(#bnfisintnorm(bnf,p)>0,l++;w=[p];v=concat(v,w))));
print("Up to 100000 there are ",j," primes congruent to 1 mod k and ",l," are norms of principal ideals")

---

(21:45) gp > read( "parinorm.txt" );
(21:47) gp > normU(29,31,100000)
Up to 100000 there are 345 primes congruent to 1 mod k and 46 are norms of principal ideals
(21:56) gp > print(v)

345/46 = 7.5, which is close to 8, the class number of field K = Q(\zeta_{29}). I would expect the same for larger n in K = Q(\zeta_{n}).

What I am curious about is how are primes p = 1 (mod n) which are NOT norms of algebraic integers in field K = Q(\zeta_{n}), represented as norms of ANY type of integer u, belonging to field.

For instance, take K = Q(\zeta_{29}), and prime p = 13457 = 1 (mod 29). There exists an ideal with norm 13457 in Q(\zeta_{29}), and in addition to that, there also exists an element u such that norm(Mod(u,polcyclo(29))) = 13457. This can be generalized to other cyclotomic fields by replacing 29 with n, and we have norm(Mod(u,polcyclo(n)). Better yet, norm(Mod(u,(x^n-1)/(x-1))) = U(n) is a cyclotomic-like divisiblity sequence with term U(29) = 13457. These are true for ideals, which are also principal.

How about primes p, which the ideals are NOT principal. Take K = Q(\zeta_{29}), and prime p = 22621 = 1 (mod 29). There are ideals with norm p = 22621 in K = Q(\zeta_{29}), however none of these ideals are principal and we do not have an algebraic integer, or element u such that norm(Mod(u,polcyclo(29)) = 22621. Is there a way to generalize the non-principal unique ideals which correspond to norm 22621 in K = Q(\zeta_{29}), to K = Q(\zeta_{n}). In other words, is it possible to come up with a sequence of non-principal ideals, and a cyclotomic divisiblity sequence with the 29th term, U(29) = 22621?

I do belive there is a way to do this, using other types of non-algebraic integers.

Any further thoughts, comments concerns? Thanks.

Last fiddled with by carpetpool on 2017-12-10 at 16:22
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