In the field of n-th 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.

? v=[];w=[];j=0;l=0;forprime(p=29,100000,if(p%46==1,j++;if(#bnfisintnorm(k,p)>0,l++;w=[p];v=concat(v,w))));print("Up to 100000 there are ",j," primes congruent to 1 mod 46 and ",l," are norms of principal ideals")

Up to 100000 there are 429 primes congruent to 1 mod 46 and 141 are norms of principal ideals