Parameterizing roots of polynomial congruences

We use the arithmetic of ideals in orders to parameterize the roots µ (mod m) of the polynomial congruence F(µ) ≡ 0 (mod m), F(X) ∈ Z[X] monic, irreducible and degree d. Our parameterization generalizes Gauss’s classic parameterization of the roots of quadratic congruences using binary quadratic forms, which had previously only been extended to the cubic polynomial F(X) = X3 − 2. We show
that only a special class of ideals are needed to parameterize the roots µ (mod m), and that in the cubic setting, d = 3, general ideals correspond to pairs of roots µ1 (mod m1), µ2 (mod m2) satisfying gcd(m1, m2, µ1 − µ2) = 1. At the end we illustrate our parameterization and this correspondence between roots and ideals with a few applications, including finding approximations to µ m ∈ R/Z, finding an explicit Euler product for the co-type zeta function of Z[2 1 3 ], and computing the composition of cubic ideals in terms of the roots µ1 (mod m1) and µ2 (mod m2).