Thanks, I will look further into these properties, I did additional tests on the fields K = Q(\zeta_{n}), for n = 31, 37, 41, 43 and found

(22:23) gp > normU(31,37,100000)

Up to 100000 there are 320 primes congruent to 1 mod k and 40 are norms of principal ideals

For n = 31, 37 they are checked up to all primes <= 100000, n = 41, 43 they are checked up to all primes <= 300000,

One question I came up with is for each prime n, what is the smallest prime which is a norm of a principal ideal in the cyclotomic field K = Q(\zeta_{n})?

Let a(n) be the smallest prime that is a norm principal ideal in the cyclotomic field K = Q(\zeta_{P(n)}) and P(n) is the nth prime,

The terms are a(n) = 3, 7, 11, 29, 23, 53, 103, 191, 599, 4931, 5953, 32783, 101107, 178021,...

For example, the a(3) = 11, because 5 is the third prime, and the smallest prime p that is a norm of a principal ideal in the cyclotomic field K = Q(\zeta_{5}) is 11.

Is there an easy method or formula for finding these terms? If someone is willing to explore further and a(15) through a(25) (the smallest primes p that are norms of principal ideals in the cyclotomic fields K = Q(\zeta_{n}), for n = 47, 53, 59, 61, 67, 73, 79, 83, 89, 97)

I considered adding an

OEIS to it, I am not sure if this is of any additional interest however.

Thanks for help.