20171019, 00:32  #23  
"Sam"
Nov 2016
325_{10} Posts 
Quote:
polcyclo(n) is the nth cyclotomic polynomial p is a norm of a principal ideal in Kn there is no principal ideal with norm p in Kn In the first case, we have elements w with norm p in Kn, we can write p as the norm of w (mod polcyclo(n)). The second case this is false. Looking at the second case, we have a prime p where there is NO principal ideal with norm p in Kn. There is (should be) a field extension Kn/Q where there are elements in Kn/Q with norm p. Like in the first case, we can (should be able to) write p as the norm of w (mod z). Here, z is a polynomial with similar properties to polcyclo(n). What I am not sure is how to generate such polynomials z, which define a specific field extension of Kn, and how to embed them in polynomial sequences for other cyclotomic field extensions. (The cyclotomic polynomials, form a sequences: 1, x+1, x^2+x+1, x^2+1, x^4+x^3+x^2+x+1, x^2x1, x^6+x^5+x^4+x^3+x^2+x+1, x^4+1, x^6+x^3+1, x^4x^3+x^2x+1,...) What about sequences like the one above, except they are sequences of polynomials z1, z2, z3,... which define a field extension. Please ask for more information, or further thoughts. Thanks. 

20171019, 14:00  #24 
Feb 2017
Nowhere
2^{3}·7·89 Posts 
I refer you to the example I gave in post #4 to this thread.

20171029, 23:47  #25 
"Sam"
Nov 2016
5^{2}·13 Posts 
In the field of the nth roots of unity (Kn), if q = p^k = 1 (mod n), q is the norm of a principal if k > 1. That is, in the field Kn, for a prime q = 1 (mod n), q^k (where k > 1) is the norm of a principal ideal. (If q^k = 1 (mod n) with k > 1, then q is not necessarily 1 (mod n) for this case to be true although in most cases it is.) For any base b > 0, b^(phi(n)) is the norm of a principal ideal.
For these two cases, let w be any element (polynomial). If the norm of w mod polcyclo(n) = m is divisible by b^(phi(n)) for all n, is m/(b^(phi(n)) the norm of a principal ideal (in the field Kn), or can it be in the nonprincipal class (in the field Kn)? If the norm of w mod polcyclo(n) = m is a perfect kth power (k > 1) for all n, is kth root of m the norm of a principal ideal (in the field Kn), or can it be in the nonprincipal class (in the field Kn)? Any help, comments, suggestions please? Thank you. 
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