In
mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many ar ...
, in the area of
abstract algebra
In mathematics, more specifically algebra, abstract algebra or modern algebra is the study of algebraic structures, which are set (mathematics), sets with specific operation (mathematics), operations acting on their elements. Algebraic structur ...
known as
Galois theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...
, the Galois group of a certain type of
field extension
In mathematics, particularly in algebra, a field extension is a pair of fields K \subseteq L, such that the operations of ''K'' are those of ''L'' restricted to ''K''. In this case, ''L'' is an extension field of ''K'' and ''K'' is a subfield of ...
is a specific
group associated with the field extension. The study of field extensions and their relationship to the
polynomial
In mathematics, a polynomial is a Expression (mathematics), mathematical expression consisting of indeterminate (variable), indeterminates (also called variable (mathematics), variables) and coefficients, that involves only the operations of addit ...
s that give rise to them via Galois groups is called
Galois theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...
, so named in honor of
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 Nth root, ...
who first discovered them.
For a more elementary discussion of Galois groups in terms of
permutation groups, see the article on
Galois theory
In mathematics, Galois theory, originally introduced by Évariste Galois, provides a connection between field (mathematics), field theory and group theory. This connection, the fundamental theorem of Galois theory, allows reducing certain problems ...
.
Definition
Suppose that
is an extension of the
field (written as
and read "''E'' over ''F''). An
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 automorphism ...
of
is defined to be an automorphism of
that fixes
pointwise. In other words, an automorphism of
is an
isomorphism
In mathematics, an isomorphism is a structure-preserving mapping or morphism 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 the ...
such that
for each
. The
set of all automorphisms of
forms a group with the operation of
function composition. This group is sometimes denoted by
If
is a
Galois extension, then
is called the Galois group of
, and is usually denoted by
.
If
is not a Galois extension, then the Galois group of
is sometimes defined as
, where
is the
Galois closure of
.
Galois group of a polynomial
Another definition of the Galois group comes from the Galois group of a polynomial