Given a polynomial function f with rational coefficients, the splitting field of f is defined to be the smallest FIELD [I.3 §2.2] that contains all rational numbers and all the roots of f. The Galois group of f is the group of all AUTOMORPHISMS [I.3 §4.1] of the splitting field. Each such automorphism permutes the roots of f, so the Galois group can be thought of as a subset of the group of all PERMUTATIONS [III.68] of these roots. The structure and properties of the Galois group are closely connected with the solubility of the polynomial: in particular, the Galois group can be used to show that not all polynomials are solvable by radicals (that is, solvable by means of a formula that involves the usual arithmetic operations together with the extraction of roots). This theorem, spectacular as it is, is by no means the only application of Galois groups: they play a central role in modern algebraic number theory.
For more details, see THE INSOLUBILITY OF THE QUINTIC [V.21] and ALGEBRAIC NUMBERS [IV.1 §20].