2020-04-18, 06:06 | #1 |
"Sam"
Nov 2016
2^{3}×41 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 |
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