Galois group
In
mathematics, 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.
Examples
- 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.
Fundamental theorem of Galois theory.
Let
E be a finite Galois extension of the field
F with Galois group
G. For every
subgroup H of
G, let
E^{H} denote the subfield of
E consisting of all elements which are fixed by all elements of
H. Then the function
- H |-> E^{H}
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/
E^{H}. Furthermore,
the field
E^{H} is a normal extension of
F if and only if
H is a
normal subgroup of
G. If
H is a normal subgroup of
G
then the restriction of
G's elements to
E^{H} induces an
isomorphism between the group
G/H and the Galois group of the extension
E^{H}/
F.