Quote:
Originally Posted by Dr Sardonicus
In the field of nth roots of unity K = Q(\zeta_{n}), the prime numbers p congruent to 1 (mod n) are precisely those which are norms of ideals in the ring of integers O_{K}. Invoking the "assumption of ignorance" that no ideal class is favored over any other, the "obvious" answer is that 1/h of these primes are norms of principal ideals (that is, are norms of algebraic integers in K), where h is the class number (standard notation).
And in fact this is correct. It may be viewed as a generalization of the equal distribution of primes in arithmetic progressions. See THE ASYMPTOTIC DISTRIBUTION OF PRIME IDEALS IN IDEAL CLASSES
For the field k = Q(\zeta _{23}) with h = 3 (constructed in advance with bnfinit()), I ran a script to get the actual count for primes up to 100,000. As you can see, it's pretty close to 1/3 of all primes congruent to 1 mod 46.

I would like to announce one result involving the field K = Q(\zeta_{23}).
**Conjecture/Observation**
For a prime p = 1 (mod 23), p is a norm of a principal ideal if and only if there are 3 solutions to x^3x1 = 0 (mod p).
To see this is the same list as the one quoted in Dr. Sardonicus's first post, I modified a PARI/GP script which includes a bnf construction of any number field, and k.
normU(k,w,n,m) =
{
bnf = bnfinit(w);
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")
here w is any polynomial, and the script finds primes n < p < m, and p = 1 (mod k) such that p is the norm of a principal ideal in the field that w defines.
The list above is the same as the one Dr. Sardonicus's first post, and also the primes p = 1 (mod 23) which satisfy x^3x1 = 0 (mod p).
In the field K = Q(\zeta_{29}), what is the polynomial w for which if p = 1 (mod 29), and p is the norm of a principal ideal, then p satisfies w = 0 (mod p), if it is not principal, then p does not satisfy w = 0 (mod p)?
Update: For K23, this works for any polynomial w having a discriminant D of 23. In fact replacing x^3x1 with x^2+x+6, x^2+3*x+8, x^3+3*x^2+2*x1,... would all give the same result.
Thanks for help.