Each of these has group structures corresponding to the operations of addition and multiplication. Everything else follows from the group axioms and the distributive rule. The goal of the theory is to say as much as possible about the roots of such equations, that is, values of x for which the equation is true. In general, the roots of the equation will not be members of the same base field as the coefficients. Looking again at any polynomial equation, one finds that it can have at most n roots in any extension field, where n is the degree of the polynomial.
T leaves the base field fixed.
Given this, not all permutations of the roots of a polynomial may be reasonable, because they don't induce an automorphism of the extension field which leaves the base field fixed. This may happen if there are polynomial relationships among the roots with coefficients in the base field. We can allow permutations that exchange x and x , or x and x , or both. But we can't allow a permutation that exchanges x and x . This set of automorphisms is actually a group, and it is called the Galois group of the extension.