Excerpted from Beachy/Blair, Abstract Algebra, 2nd Ed. © 1996

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§ 8.1 The Galois Group of a Polynomial

 
To study solvability by radicals of a polynomial equation f(x) = 0, we let K be the field generated by the coefficients of f(x), and let F be a splitting field for f(x) over K. Galois considered permutations of the roots that leave the coefficient field fixed. The modern approach is to consider the automorphisms determined by these permutations. We note that any automorphism of a field F must leave its prime subfield fixed.

 
Proposition 8.1.1. 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.

 
Definition 8.1.2. 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).

 
Definition 8.1.3. 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.

 
Proposition 8.1.4. 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.

 
Lemma 8.1.5. 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.

 
Theorem 8.1.6. 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].

 
Corollary 8.1.7. 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.

 
In Corollary 8.1.7, if we take K = Zp, where p is a prime number, and F is an extension of degree m, then the generator of the cyclic group Gal(F/K) is the automorphism : F -> F defined by (x) = xp, for all x in F. This automorphism is called the Frobenius automorphism of F. (See Example 8.3.1.)



§ 8.1 Solved problems

 
7. Determine the group of all automorphisms of a field with 4 elements.     Solution

8. Let F be the splitting field in C of x4+1.
    (i) Show that [F:Q] = 4.
    (ii) Find automorphisms of F that have fixed fields Q(), Q(i), and Q( i), respectively.
    Solution

9. Find the Galois group over Q of the polynomial x4+4.     Solution

10. Find the Galois groups of x3-2 over the fields Z5 and Z11.     Solution

11. Find the Galois group of x4-1 over the field Z7.     Solution

12. Find the Galois group of x3-2 over the field Z7.     Solution

 


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