 mersenneforum.org Prime numbers norms of modulo cyclotomic polynomials
 Register FAQ Search Today's Posts Mark Forums Read  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.   Thread Tools Show Printable Version Email this Page Similar Threads Thread Thread Starter Forum Replies Last Post Cruelty Proth Prime Search 158 2020-07-31 22:23 carpetpool Abstract Algebra & Algebraic Number Theory 0 2017-04-19 20:33 fivemack Computer Science & Computational Number Theory 2 2015-09-18 12:54 axn Computer Science & Computational Number Theory 66 2011-09-01 21:55 smslca Math 3 2011-04-18 17:18

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