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Old 2017-09-17, 13:58   #2
Dr Sardonicus
 
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Feb 2017
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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 OK. 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(\zeta23) 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

Last fiddled with by carpetpool on 2017-12-10 at 16:21
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