In mathematics, the fundamental theorem of Galois theory is a result that describes the structure of certain types of field extensions. Euclid, detail from The School of Athens by Raphael. ... In abstract algebra, a subfield of a field L is a subset K of L which is closed under the addition and multiplication operations of L and itself forms a field with these operations. ...

In its most basic form, the theorem asserts that given a field extension E/F which is finite and Galois, there is a one-to-one correspondence between its intermediate fields (fields K satisfying F ⊆ K ⊆ E; also called subextensions of E/F) and subgroups of its Galois group. In abstract algebra, an extension of a field K is a field L which contains K as a subfield. ... In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions (described below); one also says that the extension is Galois. ... In mathematics and mathematical economics, correspondence is a term with several related but not identical meanings. ... In mathematics, more specifically in abstract algebra, field extensions are the main object of study in field theory. ... In group theory, given a group G under a binary operation *, we say that some subset H of G is a subgroup of G if H also forms a group under the operation *. More precisely, H is a subgroup of G if the restriction of * to H is a group... In mathematics, a Galois group is a group associated with a certain type of field extension. ...

Proof

The proof of the fundamental theorem is not trivial. The crux in the usual treatment is a rather delicate result of Emil Artin which allows one to control the dimension of the intermediate field fixed by a given group of automorphisms. See linear independence of automorphisms of a field. Emil Artin (March 3, 1898-December 20, 1962) was an Austrian mathematician born in Vienna who spent his career in Germany (mainly in Hamburg) until the Nazi threat when he emigrated to the USA in 1937 where he was at Indiana University 1938-1946, and Princeton University 1946-1958. ...

In terms of its abstract structure, there is a Galois connection; most of its properties are fairly formal, but the actual isomorphism of the posets requires some work. In mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets (posets). Galois connections generalize the correspondence between subgroups and subfields investigated in Galois theory. ... In mathematics, a partially ordered set (or poset for short) is a set equipped with a special binary relation which formalizes the intuitive concept of an ordering. ...

Explicit description of the correspondence

For finite extensions, the correspondence can be described explicitly as follows.

• For any subgroup H of Gal(E/F), the corresponding field, usually denoted E^H, is the set of those elements of E which are fixed by every automorphism in H.
• For any intermediate field K of E/F, the corresponding subgroup is just Aut(E/K), that is, the set of those automorphisms in Gal(E/F) which fix every element of K.

For example, the topmost field E corresponds to the trivial subgroup of Gal(E/F), and the base field F corresponds to the whole group Gal(E/F).

Properties of the correspondence

The correspondence has the following useful properties.

• It is inclusion-reversing. The inclusion of subgroups H₁ ⊆ H₂ holds if and only if the inclusion of fields E^H₁ ⊇ E^H₂ holds.
• Degrees of extensions are related to orders of groups, in a manner consistent with the inclusion-reversing property. Specifically, if H is a subgroup of Gal(E/F), then |H| = [E:E^H] and [Gal(E/F):H] = [E^H:F].
• The field E^H is a normal extension of F if and only if H is a normal subgroup of Gal(E/F). In this case, the restriction of the elements of Gal(E/F) to E^H induces an isomorphism between Gal(E^H/F) and the quotient group Gal(E/F)/H.

In abstract algebra, an algebraic field extension L/K is said to be normal if L is the splitting field of a family of polynomials in K[X]. The following conditions are equivalent to L/K being a normal extension: Let Ka an algebraic closure of K containing L. Every... In mathematics, a normal subgroup N of a group G is a subgroup invariant under conjugation; that is, for each element n in N and each g in G, the element g−1ng is still in N. The statement N is a normal subgroup of G is written: . There are... In abstract algebra, a group isomorphism is a function between two groups that sets up a one-to-one correspondence between the elements of the groups in a way that respects the given group operations. ...

Example

Consider the field

Since K is first determined by adjoining √2, then √3, a typical element of K can be written

where a, b, c, d are rational numbers. Its Galois group

can be determined by examining the automorphisms of K which fix a. Each such automorphism must send √2 to either √2 or −√2, and must send √3 to either √3 or −√3. Suppose that f exchanges √2 and −√2, so

and g exchanges √3 and −√3, so

These are clearly automorphisms of K. There is also the identity automorphism e which does not change anything, and the composition of f and g which changes the signs on both radicals:

Therefore

G = {e,f,g,fg},

and G is isomorphic to the Klein four-group. It has five subgroups, each of which correspond via the theorem to a subfield of K. This article is about the mathematical group. ...

• The trivial subgroup (containing only the identity element) corresponds to all of K.
• The entire group G corresponds to the base field Q.
• The two-element subgroup {1, f} corresponds to the subfield Q(√3), since f fixes √3.
• The two-element subgroup {1, g} corresponds to the subfield Q(√2), again since g fixes √2.
• The two-element subgroup {1, fg} corresponds to the subfield Q(√6), since fg fixes √6.

Example

The following is the simplest case where the Galois group is not abelian.

Consider the splitting field K of the polynomial x³−2 over Q; that is,

where θ is a cube root of 2, and ω is a cube root of 1 (but not 1 itself). For example, if we imagine K to be inside the field of complex numbers, we may take θ to be the real cube root of 2, and ω to be

It can be shown that the Galois group

has six elements, and is isomorphic to the group of permutations of three objects. It is generated by (for example) two automorphisms, say f and g, which are determined by their effect on θ and ω,

and then

G = {1,f,f²,g,gf,gf²}.

The subgroups of G and corresponding subfields are as follows:

• As usual, the entire group G corresponds to the base field Q, and the trivial group {1} corresponds to the whole field K.
• There is a unique subgroup of order 3, namely {1, f, f²}. The corresponding subfield is Q(ω), which has degree two over Q (the minimal polynomial of ω is x² + x + 1), corresponding to the fact that the subgroup has index two in G. Also, this subgroup is normal, corresponding to the fact that the subfield is normal over Q.
• There are three subgroups of order 2, namely {1, g}, {1, gf} and {1, gf²}, corresponding respectively to the three subfields Q(θ), Q(ω²θ), Q(ωθ). These subfields have degree three over Q, again corresponding to the subgroups having index 3 in G. Note that the subgroups are not normal in G, and this corresponds to the fact that the subfields are not Galois over Q. For example, Q(θ) contains only a single root of the polynomial x³−2, so it cannot be normal over Q.

Applications

The theorem converts the difficult-sounding problem of classifying the intermediate fields of E/F into the more tractable problem of listing the subgroups of a certain finite group. In mathematics, a finite group is a group which has finitely many elements. ...

For example, to prove that the general quintic equation is not solvable by radicals (see Abel-Ruffini theorem), one first restates the problem in terms of radical extensions (extensions of the form F(α) where α is an n-th root of some element of F), and then uses the fundamental theorem to convert this statement into a problem about groups. That can then be attacked directly. Polynomial of degree 5: f(x) = (x+4)(x+2)(x+1)(x-1)(x-3)/20+2 In mathematics, a quintic equation is a polynomial equation in which the greatest exponent on the independent variable is five. ... The Abel-Ruffini theorem states that there is no general solution in radicals to polynomial equations of degree five or higher. ...

Theories such as Kummer theory and class field theory are predicated on the fundamental theorem. In mathematics, a Kummer extension of fields is a field extension L/K where for some given integer n > 1 we have [L:K] = n and L is generated over K by a root of a polynomial Xn − a with a in K, and K contains n distinct roots of... In mathematics, class field theory is a major branch of algebraic number theory. ...

Infinite case

There is also a version of the fundamental theorem that applies to infinite algebraic extensions, which are normal and separable. It involves defining a certain topological structure, the Krull topology, on the Galois group; only subgroups that are also closed sets are relevant in the correspondence. In abstract algebra, a field extension L /K is called algebraic if every element of L is algebraic over K, i. ... In abstract algebra, an algebraic field extension L/K is said to be normal if L is the splitting field of a family of polynomials in K[X]. The following conditions are equivalent to L/K being a normal extension: Let Ka an algebraic closure of K containing L. Every... In mathematics, a separable extension of a field K is a field L containing K that can be generated by adjoining to K a set of elements α, each of which is a root of a separable polynomial over K. In that case, each β in L has a separable... Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... 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. ... In topology and related branches of mathematics, a closed set is a set whose complement is open. ...