Page 295 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes 3. Any automorphism of a field f must leave its prime ................... fixed.
(a) sub group (b) sub domain

(c) sub field (d) sub range
4. Given [F : Q] is equal to ...................
(a) 7 (b) 5
(c) 6 (d) 4

5. Automorphism of F that have fixed fields Q( 2),Q(i) and Q 2,i respectively.
(a) Q  Q 2,Q  Q(i),Q 3 2i  2i
1
2
(b) Q  2 ,Q  i,Q 3 2i  0
2
1
(c) Q = Q = Q 3
2
(d) Q = Q = Q 3 –1
–1
2
1
31.6 Summary
Let F be an extension field of K. The set of all automorphisms  : F > F such that (a) = a for

all a in K is a group under composition of functions.
Let F be an extension field of K. The set

{ in Aut(F) | (a) = a for all a in K }
is called the Galois group of F over K, denoted by Gal(F/K).
Let K be a field, let f(x) be a polynomial in 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.

Let F be an extension field of K, and let f(x) be a polynomial in K[x]. Then any element of

Gal(F/K) defines a permutation of the roots of f(x) that lie in F.
Let f(x) be a polynomial in K[x] with no repeated roots and let F be a splitting field for f(x)

over K. If  : K > L is a field isomorphism that maps f(x) to g(x) in L[x] and E is a splitting
field for g(x) over L, then there exist exactly [F:K] isomorphisms : F -> E such that (a) = (a)
for all a in K.
Let K be a field, let f(x) be a polynomial in K[x], and let F be a splitting field for f(x) over

K. If f(x) has no repeated roots, then |Gal(F/K)| = [F:K].
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.
If we take K = Z , where p is a prime number, and F is an extension of degree m, then the
 p
generator of the cyclic group Gal(F/K) is the automorphism  : F -> F defined by
(x) = x , for all x in F. This automorphism is called the Frobenius automorphism of F.
p
31.7 Keywords

Galois Group: Let F be an extension field of K. The set
{ in Aut(F) | (a) = a for all a in K }
is called the Galois group of F over K, denoted by Gal(F/K).

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