2020-04-18, 06:06 | #1 |
"Sam"
Nov 2016
2×163 Posts |
Field mapping to fractional elements
Suppose we have a number field K = Q(ℽ) where ℽ is a root of the polynomial f of degree d.
Define C(f)_{n} to be the n-th coefficient of f. Suppose we have integers a and q where f(a) = 0 mod q (i.e. a is a root of f mod q, or factorization over finite field of order q if q is prime). Then define the following two polynomials: Mod[] Let N(e) be the norm of any element e ∈ O_{K}, the ring of integers in the field K. Suppose that S = R(f,ℽ)_{(a,q)} + e ∈ O_{K}, Let T be the minimal polynomial of S. Prove that T*q is a polynomial with integer coefficients (the leading coefficient is q). Suppose that N(S) = q'/q. Show that there is an element j ∈ O_{K} with N(j) = q*q'. Furthermore, is there a field mapping from S to j. That is, if we know and element j with norm N(j), can we easily find an element S (using the summation formulas above) such that N(S) = q'/q? Or if we are given S and N(S) = q'/q, find j such that N(j) = q*q'. Last fiddled with by carpetpool on 2020-04-18 at 06:12 |
Thread Tools | |
Similar Threads | ||||
Thread | Thread Starter | Forum | Replies | Last Post |
Orders of consecutive elements does not exceed floor(sqrt(p)) | carpetpool | carpetpool | 0 | 2020-03-12 02:50 |
how is it, primes in the security elements? | hal1se | Miscellaneous Math | 2 | 2018-08-30 02:06 |
Small search of cycles with odd and even elements | Drdmitry | Aliquot Sequences | 0 | 2011-12-14 13:50 |
The Elements | science_man_88 | Science & Technology | 24 | 2010-07-26 12:29 |
Fractional Calculus | nibble4bits | Math | 2 | 2008-01-11 21:46 |