View Single Post
Old 2017-10-02, 13:43   #7
Dr Sardonicus
Dr Sardonicus's Avatar
Feb 2017

132·23 Posts

Determining the least prime in an arithmetic progression is already known to be a very hard problem. The classic result is due to Kulik. In a sense it is satisfying (given the difficulty of the problem), but it is far from what is suspected.

What might be more interesting here, is, for a given n, noting the class number h = h(n) for the field of n-th roots of unity, and the number k of primes congruent to 1 (mod n) you have to go through until you find one that is the norm of principal ideal. My guess is, k will not be terribly larger than h, but it could be much smaller.

BTW, if n is a prime power qf, then 1 - \zeta_{n} is a principal generator of the (unique) prime ideal of norm q.

Last fiddled with by Dr Sardonicus on 2017-10-02 at 14:17
Dr Sardonicus is offline   Reply With Quote