In mathematics, a Galois group is a group associated with a certain type of field extension. The study of field extensions (and polynomials which give rise to them) via Galois groups is called Galois theory. History Main article: History of mathematics In addition to recognizing how to count concrete objects, prehistoric peoples also recognized how to count abstract quantities, like time -- days, seasons, years. ... In mathematics, a group is a set, together with a binary operation, such as multiplication or addition, satisfying certain axioms, detailed below. ... In abstract algebra, an extension of a field K is a field L which contains K as a subfield. ... In mathematics, polynomial functions, or polynomials, are an important class of simple and smooth functions. ... In mathematics, Galois theory is that branch of abstract algebra which studies the symmetries of the roots of polynomials. ...

For a more elementary discussion of Galois groups in terms of permutation groups, see the article on Galois theory. In mathematics, Galois theory is that branch of abstract algebra which studies the symmetries of the roots of polynomials. ...

Definition of the Galois group

Suppose that E is an extension of the field F. Consider the set of all field automorphisms of E/F; that is, isomorphisms α from E to itself, such that α(x) = x for every x in F. This set of automorphisms with the operation of function composition forms a group G, sometimes denoted Aut(E/F). In abstract algebra, an extension of a field K is a field L which contains K as a subfield. ... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication, and division (except division by zero) may be performed and the associative, commutative, and distributive rules hold, which are familiar from the arithmetic of ordinary numbers. ... In mathematics, an automorphism is an isomorphism from a mathematical object to itself. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. ... In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ...

If E/F is a Galois extension, then G is called the Galois group of the extension, and is usually denoted Gal(E/F). The significance of an extension being Galois is that it obeys the fundamental theorem of Galois theory. In mathematics, a Galois extension is a field extension that has a Galois group. ... In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions. ...

It can be shown that E is algebraic over F if and only if the Galois group is pro-finite. In abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i. ... In mathematics, pro-finite groups are groups that are in a certain sense assembled from finite groups; they share many properties with the finite groups. ...

Examples

• If E = F, then the Galois group is the trivial group that has a single element.
• If F is the field R of real numbers, and E is the field C of complex numbers, then the Galois group has two elements, namely the identity automorphism and the complex conjugation automorphism.
• If F is Q (the field of rational numbers), and E is Q(√2), the field obtained from Q by adjoining √2, then the Galois group again has two elements: the identity automorphism, and the automorphism which exchanges √2 and -√2.
• If F is Q, and E is Q(α), where α is the real cube root of 2, then E/F is not a Galois extension. This is because it is not a normal extension, since the other two cube roots of 2, being complex numbers, are not contained in Q(α). In other words E is not a splitting field. There is no automorphism of E apart from the identity.
• If F is Q and E is the field of real numbers, then the Galois group is infinite.