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2017-10-19, 00:32   #23
carpetpool

"Sam"
Nov 2016

32510 Posts

Quote:
 Originally Posted by Dr Sardonicus Oops, slight oopsadaisy I shouldn't have excluded the value n = 2. If n is even, you need an n-th root of -p rather than of p. This also assumes x^n + p or x^n - p is irreducible over the field of n-th roots of unity. Alas, that isn't always true (e.g. p = 2, n = 8). If n is even, any extension defined by polynomial with constant term p which is irreducible in K[x] will fill the bill. if n is odd, an irreducible polynomial with constant term -p instead of p will do the job.
Let Kn be the the nth cyclotomic field (field of the nth roots of unity). For some Kn the class number h = 1. Most times, h > 1. In the case that h > 1, primes p = 1 (mod n) can be classified into two important categories:

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^2-x-1, x^6+x^5+x^4+x^3+x^2+x+1, x^4+1, x^6+x^3+1, x^4-x^3+x^2-x+1,...)

What about sequences like the one above, except they are sequences of polynomials z1, z2, z3,... which define a field extension.

Thanks.

2017-10-19, 14:00   #24
Dr Sardonicus

Feb 2017
Nowhere

23·7·89 Posts

Quote:
 Originally Posted by carpetpool 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.
I refer you to the example I gave in post #4 to this thread.

 2017-10-29, 23:47 #25 carpetpool     "Sam" Nov 2016 52·13 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 non-principal 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 non-principal class (in the field Kn)? Any help, comments, suggestions please? Thank you.

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