Galois group
When you extend a field, the extra elements often come with a hidden symmetry: there may be several ways to rearrange them that respect all the arithmetic and leave the original field perfectly untouched. The Galois group collects all those symmetries of the extension into a single group, so the geometry of the extension becomes the algebra of a group.
Precisely, for a field extension L/K, the Galois group, written Gal(L/K), is the group of all automorphisms of L that fix every element of K. Each such automorphism permutes the roots of polynomials, but never disturbs anything in K. Under composition these automorphisms form a group, with the identity map as the identity element.
The Galois group is most powerful when L is a Galois extension of K — meaning both normal and separable. In that case the group's size equals the degree of the extension, and the group encodes precisely how the roots can be swapped consistently. This is the bridge that turns a question about an equation into a question about a finite group.
The Galois group of Q(sqrt(2))/Q has exactly two elements: the identity, and the map that sends sqrt(2) to -sqrt(2) (fixing every rational). It is the cyclic group of order 2.
Gal(Q(sqrt(2))/Q) is cyclic of order 2.
The group is named for Évariste Galois (1811–1832), who introduced these symmetry groups as a teenager and died in a duel at twenty. Whether a polynomial is solvable by radicals depends entirely on the structure of its Galois group.