Fundamental theorem of Galois theory
Encyclopedia
In mathematics
, the fundamental theorem of Galois theory
is a result that describes the structure of certain types of field extension
s.
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 and subgroup
s of its Galois group
. (Intermediate fields are fields K satisfying F ⊆ K ⊆ E; they are also called subextensions of E /F.)
which allows one to control the dimension of the intermediate field fixed by a given group of automorphisms. The automorphisms of a Galois extension K/F are linearly independent as functions over the field K. The proof of this fact follows from a more general notion, namely, the linear independence of characters.
There is also a fairly simple proof using the primitive element theorem
. This proof seems to be ignored by most modern treatments, possibly because it requires a separate (but easier) proof in the case of finite fields.
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.
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 ).
where a, b, c, d are rational numbers. Its Galois group G = Gal (K/Q) 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 since the permutations in a Galois group can only permute the roots of an irreducible polynomial. 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
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.
Consider the splitting field
K of the polynomial x3−2 over Q; that is, K = Q (θ, ω),
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 G = Gal (K/Q) 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
The subgroups of G and corresponding subfields are as follows:
.
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.
Theories such as Kummer theory
and class field theory
are predicated on the fundamental theorem.
s, 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 set
s are relevant in the correspondence.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the fundamental theorem of Galois theory
Galois theory
In mathematics, more specifically in abstract algebra, Galois theory, named after Évariste Galois, provides a connection between field theory and group theory...
is a result that describes the structure of certain types of field extension
Field extension
In abstract algebra, field extensions are the main object of study in field theory. The general idea is to start with a base field and construct in some manner a larger field which contains the base field and satisfies additional properties...
s.
In its most basic form, the theorem asserts that given a field extension E /F which is finite and Galois
Galois extension
In mathematics, a Galois extension is an algebraic field extension E/F satisfying certain conditions ; one also says that the extension is Galois. The significance of being a Galois extension is that the extension has a Galois group and obeys the fundamental theorem of Galois theory.The definition...
, there is a one-to-one correspondence
Correspondence (mathematics)
In mathematics and mathematical economics, correspondence is a term with several related but not identical meanings.* In general mathematics, correspondence is an alternative term for a relation between two sets...
between its intermediate fields and subgroup
Subgroup
In group theory, given a group G under a binary operation *, a subset H of G is called 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 x H is a group operation on H...
s of its Galois group
Galois group
In mathematics, more specifically in the area of modern algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension...
. (Intermediate fields are fields K satisfying F ⊆ K ⊆ E; they are also called subextensions of E /F.)
Proof
The proof of the fundamental theorem is not trivial. The crux in the usual treatment is a rather delicate result of Emil ArtinEmil Artin
Emil Artin was an Austrian-American mathematician of Armenian descent.-Parents:Emil Artin was born in Vienna to parents Emma Maria, née Laura , a soubrette on the operetta stages of Austria and Germany, and Emil Hadochadus Maria Artin, Austrian-born of Armenian descent...
which allows one to control the dimension of the intermediate field fixed by a given group of automorphisms. The automorphisms of a Galois extension K/F are linearly independent as functions over the field K. The proof of this fact follows from a more general notion, namely, the linear independence of characters.
There is also a fairly simple proof using the primitive element theorem
Primitive element theorem
In mathematics, more specifically in the area of modern algebra known as field theory, the primitive element theorem or Artin's theorem on primitive elements is a result characterizing the finite degree field extensions that possess a primitive element...
. This proof seems to be ignored by most modern treatments, possibly because it requires a separate (but easier) proof in the case of finite fields.
In terms of its abstract structure, there is a Galois connection
Galois connection
In mathematics, especially in order theory, a Galois connection is a particular correspondence between two partially ordered sets . The same notion can also be defined on preordered sets or classes; this article presents the common case of posets. Galois connections generalize the correspondence...
; most of its properties are fairly formal, but the actual isomorphism of the posets requires some work.
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 EH, 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 EH1 ⊇ EH2 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:EH] and [Gal(E /F ):H ] = [EH:F ].
- The field EH is a normal extensionNormal extensionIn 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]...
of F (or, equivalently, Galois extension, since any subextension of a separable extension is separable) if and only if H is a normal subgroupNormal subgroupIn abstract algebra, a normal subgroup is a subgroup which is invariant under conjugation by members of the group. Normal subgroups can be used to construct quotient groups from a given group....
of Gal(E /F ). In this case, the restriction of the elements of Gal(E /F ) to EH induces an isomorphismGroup isomorphismIn 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. If there exists an isomorphism between two groups, then the groups are called isomorphic...
between Gal(EH/F ) and the quotient groupQuotient groupIn mathematics, specifically group theory, a quotient group is a group obtained by identifying together elements of a larger group using an equivalence relation...
Gal(E /F )/H.
Example
Consider the field K = Q(√2, √3) = Q(√2)(√3). Since K is first determined by adjoining √2, then √3, each element of K can be written as:where a, b, c, d are rational numbers. Its Galois group G = Gal (K/Q) 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 since the permutations in a Galois group can only permute the roots of an irreducible polynomial. 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
and G is isomorphic to the Klein four-group
Klein four-group
In mathematics, the Klein four-group is the group Z2 × Z2, the direct product of two copies of the cyclic group of order 2...
. It has five subgroups, each of which correspond via the theorem to a subfield of K.
- 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
Splitting field
In abstract algebra, a splitting field of a polynomial with coefficients in a field is a smallest field extension of that field over which the polynomial factors into linear factors.-Definition:...
K of the polynomial x3−2 over Q; that is, K = Q (θ, ω),
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 G = Gal (K/Q) 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
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, f 2}. 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 indexIndex of a subgroupIn mathematics, specifically group theory, the index of a subgroup H in a group G is the "relative size" of H in G: equivalently, the number of "copies" of H that fill up G. For example, if H has index 2 in G, then intuitively "half" of the elements of G lie in H...
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, gf 2}, corresponding respectively to the three subfields Q(θ), Q(ωθ), Q(ω2θ). 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 groupFinite group
In mathematics and abstract algebra, a finite group is a group whose underlying set G has finitely many elements. During the twentieth century, mathematicians investigated certain aspects of the theory of finite groups in great depth, especially the local theory of finite groups, and the theory of...
.
For example, to prove that the general quintic equation is not solvable by radicals (see Abel–Ruffini theorem
Abel–Ruffini theorem
In algebra, the Abel–Ruffini theorem states that there is no general algebraic solution—that is, solution in radicals— to polynomial equations of degree five or higher.- Interpretation :...
), 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.
Theories such as Kummer theory
Kummer theory
In abstract algebra and number theory, Kummer theory provides a description of certain types of field extensions involving the adjunction of nth roots of elements of the base field. The theory was originally developed by Ernst Eduard Kummer around the 1840s in his pioneering work on Fermat's last...
and class field theory
Class field theory
In mathematics, class field theory is a major branch of algebraic number theory that studies abelian extensions of number fields.Most of the central results in this area were proved in the period between 1900 and 1950...
are predicated on the fundamental theorem.
Infinite case
There is also a version of the fundamental theorem that applies to infinite algebraic extensionAlgebraic extension
In abstract algebra, a field extension L/K is called algebraic if every element of L is algebraic over K, i.e. if every element of L is a root of some non-zero polynomial with coefficients in K. Field extensions that are not algebraic, i.e...
s, which are normal
Normal extension
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]...
and separable
Separable extension
In modern algebra, an algebraic field extension E\supseteq F is a separable extension if and only if for every \alpha\in E, the minimal polynomial of \alpha over F is a separable polynomial . Otherwise, the extension is called inseparable...
. It involves defining a certain topological structure, the Krull topology, on the Galois group; only subgroups that are also closed set
Closed set
In geometry, topology, and related branches of mathematics, a closed set is a set whose complement is an open set. In a topological space, a closed set can be defined as a set which contains all its limit points...
s are relevant in the correspondence.