Cyclotomic Splitting Fields
Another idea on the cyclotomic fields with class number > 1 is that K = Q(\zeta_{n}) is split into some number of separate fields. Each prime p = 1 (mod n) falls into the category of one of these splitting fields (ideal classes), and can be expressed as the norm of a "principal ideal" in that splitting field.
In K23 = Q(\zeta_{23}) the primes p = 1 (mod 23) are split into two different ideal classes depending on weather p or 47*p is a norm of a principal ideal.
In K29 = Q(\zeta_{29}) the primes p = 1 (mod 29) are split into two different ideal classes depending on weather p or 59*p is a norm of a principal ideal.
In K31 = Q(\zeta_{31}) the primes p = 1 (mod 31) are split into three different ideal classes depending on weather p, 311*p, or 1117*p is a norm of a principal ideal.
In K37 = Q(\zeta_{37}) the primes p = 1 (mod 37) are split into two different ideal classes depending on weather p, or 149*p is a norm of a principal ideal.
What are the cases for Kn = Q(\zeta_{n}) (primes n = 41, 43, 47)?
Are my assertions correct for Kn (n = 23, 29, 31, 37)?
Thanks for help.
Last fiddled with by carpetpool on 20171002 at 03:47
