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**Proposition 8.3.1.**
Let F be a field, and let G be a subgroup of Aut(F). Then

{ a in F | (a) = a for all in G }

is a subfield of F.

**Definition 8.3.2.**
Let F be a field, and let G be a subgroup of Aut (F). Then

{ a in F | (a) = a for all in G }

is called the

**Proposition 8.3.3.**
If F is the splitting field over K of a separable polynomial and G = Gal(F/K),
then F^{G} = K.

**Lemma 8.3.4. [Artin]**
Let G be a finite group of automorphisms of the field F,
and let K = F^{G}. Then

[F:K] | G |.

**Definition 8.3.5.**
Let F be an algebraic extension of the field K.
Then F is said to be a
** normal **
extension of K if every irreducible polynomial in K[x] that
contains a root in F is a product of linear factors in F[x].

**Theorem 8.3.6.**
The following conditions are equivalent for an extension field F of K:

**(1)**F is the splitting field over K of a separable polynomial;**(2)**K = F^{G}for some finite group G of automorphisms of F;**(3)**F is a finite, normal, separable extension of K.

**Corollary 8.3.7.**
If F is an extension field of K such that K = F^{G}
for some finite group G of automorphisms of F,
then G = Gal(F/K).

**Example** 8.3.1.
The Galois group of GF(p^{n})
over GF(p) is cyclic of order n,
generated by the automorphism
defined by
(x) = x^{p},
for all x in GF(p^{n}).
This automorphism is usually known as the
**Frobenius automorphism**
of GF(p^{n}).

**Theorem 8.3.8. [Fundamental Theorem of Galois Theory]**
Let F be the splitting field of a separable polynomial over the field K,
and let G = Gal(F/K).

**(a)**There is a one-to-one order-reversing correspondence between subgroups of G and subfields of F that contain K:- (i) If H is a subgroup of G, then the corresponding subfield is
F
^{H}, andH = Gal(F/F

^{H}). (ii) If E is a subfield of F that contains K, then the corresponding subgroup of G is H = Gal(F/E), and

E = F

^{H}.

- (i) If H is a subgroup of G, then the corresponding subfield is
F
**(b)**For any subgroup H of G, we have[F:F

^{H}] = | H | and [F^{H}:K] = [G:H].**(c)**Under the above correspondence, the subgroup H is normal if and only if the subfield E = F^{H}is a normal extension of K. In this case,Gal(E/K) Gal(F/K) / Gal(F/E).

In the statement of the fundamental theorem we could have simply
said that normal subgroups correspond to normal extensions.
In the proof we noted that if E is a normal extension of K, then
(E)
E for all
in Gal(F/K).
In the context of the fundamental theorem,
we say that two intermediate subfields
E_{1} and E_{2} are
**conjugate**
if there exists
in Gal(F/K) such that
( E_{1} ) = E_{2}.
The next result shows that the subfields conjugate to an intermediate
subfield E correspond to the subgroups conjugate to Gal(F/E).
Thus E is a normal extension if and only if it is conjugate only to itself.

**Proposition 8.3.9.**
Let F be the splitting field of a separable polynomial over the field K,
and let E be a subfield such that
K E
F,
with H = Gal(F/E).
If
is in Gal(F/K), then

Gal(F/(E)) =
H
^{-1}.

**Theorem 8.3.10. [Fundamental Theorem of Algebra]**
Any polynomial in **C**[x] has a root in **C**.

**6.**
Prove that if F is a field extension of K
and K = F^{G} for a finite group G of automorphisms of F,
then there are only finitely many subfields between F and K.
*Solution*

**7.**
Let F be the splitting field over K of a separable polynomial.
Prove that if Gal (F/K) is cyclic,
then for each divisor d of [F:K] there is
exactly one field E with
K
E F and [E:K] = d.
*Solution*

**8.**
Let F be a finite, normal extension of **Q**
for which | Gal (F/**Q**) | = 8
and each element of Gal (F/**Q**) has order 2.
Find the number of subfields of F that have degree 4 over **Q**.
*Solution*

**9.**
Let F be a finite, normal, separable extension of the field K.
Suppose that the Galois group Gal (F/K) is isomorphic to D_{7}.
Find the number of distinct subfields between F and K.
How many of these are normal extensions of K?
*Solution*

**10.**
Show that
F = **Q**(, i)
is normal over **Q**;
find its Galois group over **Q**,
and find all intermediate fields between **Q** and F.
*Solution*

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