Automorphism of E
E is an extension of field F.
A ring isomorphism from E onto E.
Galois Group of E Over F
The set of all automorphisms of E that take every
element of F to itself.
Denoted Gal (E / F ) .
Fixed field of H
H is a subgroup of Gal (E / F ) .
The set EH  {x  E |  ( x)  x  H } .
Fundamental Theorem of Galois Theory
(Part 1 of 2)
Let F be a field of characteristic 0 or a finite field. If E is
the splitting field over F for some polynomial in F [ x] ,
then the mapping from the set of subfields of E
containing F to the set of subgroups of Gal (E / F )
given by K  Gal (E / K ) is a one-to-one
correspondence. Furthermore, for any subfield K of E
containing F,
1. [E : K ] | Gal (E / K )| and
[K : F ] | Gal(E / F )| / | Gal(E / K )| . [The index of
Gal (E / K ) in Gal(E / F ) equals the degree of K
over F.]
Fundamental Theorem of Galois Theory
(Part 2 of 2)
2. If K is the splitting field of some polynomial in
F [ x] , then Gal (E / K ) is a normal subgroup of
Gal(E / F ) and Gal (K / F ) is isomorphic to
Gal(E / F ) / Gal(E / K ) .
3. K  EGal ( E / K ) . [The fixed field of Gal (E / K ) is
K.]
4. If H is a subgroup of Gal (E / F ) , then
H  Gal ( E / EH ) . [The automorphism group of E
fixing EH is H.]
Solvable by Radicals Over F
Let F be a field, and f ( x)  F[ x] .
f ( x) splits in some extension F (a1, a2 ,..., an ) of F and
there exist positive integers k1, k2 ,..., kn such that
a1k1  F and aiki  F (a1,..., ai1) for i  2,..., n .
Solvable Group
A group G has a series of subgroups
{e}  H 0  H1  ...  H k  G , where for each 0  i  k ,
H i is normal in Hi1 and H i1 / H i is Abelian.
Splitting Field of xn  a
Let F be a field of characteristic 0 and let a  F . If E is
the splitting field of xn  a over F, then the Galois group
Gal(E / F ) is solvable.
Factor Group of a Solvable Group Is Solvable
A factor group of a solvable group is solvable.
N and G/N Solvable Implies G Is Solvable
Let N be a normal subgroup of a group G. If both N and
G/N are solvable, then G is solvable.
Solvable by Radicals Implies Solvable Group
Let F be a field of characteristic 0 and let f ( x)  F[ x] .
Suppose that f ( x) splits in F (a1, a2 ,..., at ) , where
a1n1  F and aini  F (a1,..., ai1) for i  2,..., t . Let E be
the splitting field for f ( x) over F in F (a1, a2 ,..., an ) .
Then the Galois group Gal (E / F ) is solvable.