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'.