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 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/EH. Furthermore,
the field EH 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 EH induces an
isomorphism between the group G/H and the Galois group of the extension
EH/F.
