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carpetpool 2020-04-18 06:06

Field mapping to fractional elements
 
Suppose we have a number field [B]K[/B] = Q(ℽ) where ℽ is a root of the polynomial f of degree d.

Define C(f)[SUB]n[/SUB] 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:

[TEX] M(f,x)_l = \sum_{i=1}^{l} { C(f)_i x^{l-i} } [/TEX]

[TEX] R(f,x)_(a,q) = \sum_{i=1}^{d} [/TEX] Mod[[TEX]({ M(f,x)_i x^{d-i}}, q)[/TEX]]

Let N(e) be the norm of any element e ∈ O[SUB]K[/SUB], the ring of integers in the field [B]K[/B].

Suppose that S = R(f,ℽ)[SUB](a,q)[/SUB] + e ∈ O[SUB]K[/SUB],

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[SUB]K[/SUB] 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'.


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