In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the fundamental theorem of Galois theory is a result that describes the structure of certain types of
field extension
In mathematics, particularly in algebra, a field extension is a pair of fields E\subseteq F, such that the operations of ''E'' are those of ''F'' restricted to ''E''. In this case, ''F'' is an extension field of ''E'' and ''E'' is a subfield of ...
s in relation to
groups. It was proved by
Évariste Galois
Évariste Galois (; ; 25 October 1811 – 31 May 1832) was a French mathematician and political activist. While still in his teens, he was able to determine a necessary and sufficient condition for a polynomial to be solvable by radical ...
in his development of
Galois theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems in field theory to ...
.
In its most basic form, the theorem asserts that given a field extension ''E''/''F'' that is
finite
Finite is the opposite of infinite. It may refer to:
* Finite number (disambiguation)
* Finite set, a set whose cardinality (number of elements) is some natural number
* Finite verb, a verb form that has a subject, usually being inflected or marke ...
and
Galois, there is a
one-to-one correspondence
In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other ...
between its intermediate fields and
subgroup
In group theory, a branch of mathematics, 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 ...
s of its
Galois group
In mathematics, in the area of abstract algebra known as Galois theory, the Galois group of a certain type of field extension is a specific group associated with the field extension. The study of field extensions and their relationship to the po ...
. (''Intermediate fields'' are
fields ''K'' satisfying ''F'' ⊆ ''K'' ⊆ ''E''; they are also called ''subextensions'' of ''E''/''F''.)
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
fixed field, denoted ''E
H'', is the
set
Set, The Set, SET or SETS may refer to:
Science, technology, and mathematics Mathematics
*Set (mathematics), a collection of elements
*Category of sets, the category whose objects and morphisms are sets and total functions, respectively
Electro ...
of those elements of ''E'' which are fixed by every
automorphism
In mathematics, an automorphism is an isomorphism from a mathematical object to itself. It is, in some sense, a symmetry of the object, and a way of mapping the object to itself while preserving all of its structure. The set of all automorphis ...
in ''H''.
* For any intermediate field ''K'' of ''E''/''F'', the corresponding subgroup is Aut(''E''/''K''), that is, the set of those automorphisms in Gal(''E''/''F'') which fix every element of ''K''.
The fundamental theorem says that this correspondence is a one-to-one correspondence if (and only if) ''E''/''F'' is a
Galois extension
In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ' ...
.
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'').
The notation Gal(''E''/''F'') is only used for
Galois extension
In mathematics, a Galois extension is an algebraic field extension ''E''/''F'' that is normal and separable; or equivalently, ''E''/''F'' is algebraic, and the field fixed by the automorphism group Aut(''E''/''F'') is precisely the base field ' ...
s. If ''E''/''F'' is Galois, then Gal(''E''/''F'') = Aut(''E''/''F''). If ''E''/''F'' is not Galois, then the "correspondence" gives only an
injective
In mathematics, an injective function (also known as injection, or one-to-one function) is a function that maps distinct elements of its domain to distinct elements; that is, implies . (Equivalently, implies in the equivalent contrapositi ...
(but not
surjective
In mathematics, a surjective function (also known as surjection, or onto function) is a function that every element can be mapped from element so that . In other words, every element of the function's codomain is the image of one element o ...
) map from
to
, and a surjective (but not injective) map in the reverse direction. In particular, if ''E''/''F'' is not Galois, then ''F'' is not the fixed field of any subgroup of Aut(''E''/''F'').
Properties of the correspondence
The correspondence has the following useful properties.
* It is ''inclusion-reversing''. The inclusion of subgroups ''H''
1 ⊆ ''H''
2 holds if and only if the inclusion of fields ''E''
''H''1 ⊇ ''E''
''H''2 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'', =
H''">'E'':''EH''and , Gal(''E''/''F''), /, ''H'', =
H'':''F''">'EH'':''F''
* The field ''E
H'' is a
normal extension
In abstract algebra, a normal extension is an algebraic field extension ''L''/''K'' for which every irreducible polynomial over ''K'' which has a root in ''L'', splits into linear factors in ''L''. These are one of the conditions for algebraic e ...
of ''F'' (or, equivalently, Galois extension, since any subextension of a separable extension is separable) if and only if ''H'' is a
normal subgroup
In abstract algebra, a normal subgroup (also known as an invariant subgroup or self-conjugate subgroup) is a subgroup that is invariant under conjugation by members of the group of which it is a part. In other words, a subgroup N of the group G ...
of Gal(''E''/''F''). In this case, the restriction of the elements of Gal(''E''/''F'') to ''E
H'' induces an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
between Gal(''E
H''/''F'') and the
quotient group
A quotient group or factor group is a mathematical group obtained by aggregating similar elements of a larger group using an equivalence relation that preserves some of the group structure (the rest of the structure is "factored" out). For exam ...
Gal(''E''/''F'')/''H''.
Example 1
Consider the field
:
Since is constructed from the base field
by adjoining , then , each element of can be written as:
:
Its Galois group
comprises the automorphisms of which fix . Such automorphisms must send to or , and send to or , since they permute the roots of any irreducible polynomial. Suppose that exchanges and , so
:
and exchanges and , so
:
These are clearly automorphisms of , respecting its addition and multiplication. There is also the identity automorphism which fixes each element, and the composition of and which changes the signs on ''both'' radicals:
:
Since the order of the Galois group is equal to the degree of the field extension,
, there can be no further automorphisms:
:
which is isomorphic to the
Klein four-group
In mathematics, the Klein four-group is a group with four elements, in which each element is self-inverse (composing it with itself produces the identity)
and in which composing any two of the three non-identity elements produces the third one ...
. Its five subgroups correspond to the fields intermediate between the base
and the extension .
* The trivial subgroup corresponds to the entire extension field .
* The entire group corresponds to the base field
* The subgroup corresponds to the subfield
since fixes .
* The subgroup corresponds to the subfield
since fixes .
* The subgroup corresponds to the subfield
since fixes .
Example 2
The following is the simplest case where the Galois group is not abelian.
Consider the
splitting field
In abstract algebra, a splitting field of a polynomial with coefficients in a field is the smallest field extension of that field over which the polynomial ''splits'', i.e., decomposes into linear factors.
Definition
A splitting field of a poly ...
''K'' of the
irreducible polynomial over
; that is,
where ''θ'' is a cube root of 2, and ''ω'' is a cube root of 1 (but not 1 itself). If we consider ''K'' inside the complex numbers, we may take