the galois group of a polynomial is the set of all transformations of the splitting field which preserve the ground field and the roots of the polynomial. (in mathematical jargon, these transformations are called automorphisms.) the galois group of consists of two elements: the identity transformation, which sends every complex number to itself, and complex conjugation, which sends +"i" to −"i". since the galois group does not change the ground field, it leaves the coefficients of the polynomial unchanged, so it must leave the set of all roots unchanged. each root can move to another root, however, so transformation determines a permutation of the "n" roots among themselves. the significance of the galois group derives from the fundamental theorem of galois theory, which proves that the fields lying between the ground field and the splitting field are in one-to-one correspondence with the subgroups of the galois group.