information:classification, information:extraction
Polynomials

Given an equation in unknown x

with coefficients in a field K, one can equivalently say that the solutions of (E) in K are the roots in K of the polynomial

It can be shown that a polynomial of degree n in a field has at most n roots. The equation (E) therefore has at most n solutions.

If K' is a field extension of K, one may consider (E) to be an equation with coefficients in K and the solutions of (E) in K are also solutions in K' (the converse does not hold in general). It is always possible to find a field extension of K known as the rupture field of the polynomial P, in which (E) has at least one solution.