Page 274 - DMTH403_ABSTRACT_ALGEBRA

Unit 28: Galois Theory

2. Let F be extension field of K. The set { Q  Aut(F) | Q(a) = a for all a  K } is Galois group Notes
is denoted by ................
(a) Gal(F/K) (b) Gal(u/F)
(c) Gal-1(K/F) (d) Gal(k × F)
3. Let K be a finite field and let F be an extension of K with [F : k] = m. Then Gal(F/k) is a
................ group of order m.
(a) cyclic (b) polynomial
(c) permutation (d) finite

4. A polynomial f(x) over the field k is called ................ if its irreducible factors have only
simple roots.
(a) spittery field (b) extension field

(c) separable (d) finite field
5. The ................ F of K is called simple extensions. If then exist an element u  F. Such that
F = K(u).

(a) finite field (b) extension field
(c) separable field (d) spliting field

28.4 Summary

Let F be an extension field of K. The set

{   Aut(F) | (a) = a for all a  K }

is called the Galois group of F over K, denoted by Gal(F/K).
Let K be a field, let f(x)  K[x], and let F be a splitting field for f(x) over K. Then Gal(F/K)

is called the Galois group of f(x) over K, or the Galois group of the equation f(x) = 0 over
K.
It states that if F is an extension field of K, and f(x)  K[x], then any element of Gal(F/K)

defines a permutation of the roots of f(x) that lie in F. The next theorem is extremely
important.

Let K be a field, let f(x)  K[x] have positive degree, and let F be a splitting field for f(x)

over K. If no irreducible factor of f(x) has repeated roots, then j Gal(F=K)j = [F : K].
This result can be used to compute the Galois group of any finite extension of any finite
field, but first we need to review the structure of finite fields. If F is a finite field of
characteristic p, then it is a vector space over its prime subfield Z , and so it has p elements,
n
p
where [F : Z ] = n. The structure of F is determined by the following theorem.
p
If F is a finite field with p elements, then F is the splitting field of the polynomial x p n – x
n

over the prime subfield of F.
Let K be a finite field and let F be an extension of K with [F : K] = m. Then Gal(F/K) is a

cyclic group of order m.
(The 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).

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