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Old 2020-04-18, 06:06   #1
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Nov 2016

14816 Posts
Post 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:

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

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

Let N(e) be the norm of any element e ∈ OK, the ring of integers in the field K.

Suppose that S = R(f,ℽ)(a,q) + e ∈ OK,

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 ∈ OK 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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