, 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
Suppose E is an extension of the field F, and consider the set of all
field automorphisms of E which fix F pointwise.
This set of automorphisms forms a group G.
If there are no elements of E \\ F which are fixed by all members of G,
then the extension E/F is called a Galois extension,
and G is the Galois group of the extension and is usually denoted Gal(E/F).
It can be shown that E is algebraic
over F if and only if the Galois group is pro-finite.
Fundamental theorem of Galois theory.
- If E = F, then the Galois group is the trivial group that has a single element.
- If F is the field of real numbers, and E is the field of complex numbers, then the Galois group has 2 elements.
- If F is Q (the field of rational numbers), and E is Q(√2), then the Galois group again has 2 elements.
- If F is Q, and E is Q(the real cube root of 2), then the Galois group has 1 element. (This is because the other two cube roots of 2 are complex.)
- If F is Q and E is the real numbers, then the Galois group has 1 element.
be a finite Galois extension of the field F
with Galois group
. For every subgroup H
, let EH
denote the subfield of E
consisting of all elements which are fixed by all elements of H
. Then the function
- H |-> EH
is a bijection between the set of subgroups of G
and the set of subfields of E
that contain F
. This function is monotone decreasing
and its inverse is given by the Galois group of E
the field EH
is a normal extension of F
if and only if
is a normal subgroup
. If H
is a normal subgroup of G
then the restriction of G
's elements to EH
between the group G/H
and the Galois group of the extension