Page 275 - DMTH403_ABSTRACT_ALGEBRA
P. 275
Abstract Algebra
Notes 28.5 Keywords
Prime Subfield: If F is a finite field with p elements, then F is the splitting field of the polynomial
n
x p n x over the prime subfield of F.
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).
28.6 Review Questions
1. Determine the group of all automorphisms of a field with 4 elements.
2. Let F be the splitting field in C of x + 1.
4
(a) Show that [F : Q] = 4.
(b) Find automorphisms of F that have fixed fields Q( 2 ), Q(i), and Q( 2 i),
respectively.
3. Find the Galois group over Q of the polynomial x + 4.
4
4. Find the Galois groups of x 2 over the fields Z and Z .
3
11
5
5. Find the Galois group of x 1 over the field Z .
4
7
6. Find the Galois group of x 2 over the field Z .
3
7
7. Let f(x) 2 Q[x] be irreducible over Q, and let F be the splitting field for f(x) over Q. If [F : Q]
is odd, prove that all of the roots of f(x) are real.
8. Find an element with Q( 2 , i) = Q().
9. Find the Galois group of x 1 over Z .
6
7
10. Prove that if F is a field and K = F for a finite group G of automorphisms of F, then there
G
are only finitely many subfields between F and K.
11. 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.
12. 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.
13. Let F be a finite, normal, separable extension of the field K. Suppose that the Galois group
Gal(F/K) is isomorphic to D . Find the number of distinct subfields between F and K. How
7
many of these are normal extensions of K?
14. Show that F = Q(i, 2 ) is normal over Q; find its Galois group over Q, and find all
intermediate fields between Q and F.
15. Let F = Q( 2 , 2 ). Find [F : Q] and prove that F is not normal over Q.
3
16. Find the order of the Galois group of x 2 over Q.
5
Answers: Self Assessment
1. (b) 2. (a) 3. (a) 4. (c) 5. (b)
268 LOVELY PROFESSIONAL UNIVERSITY
