Page 285 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes  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
[Fundamental Theorem of Algebra] Any polynomial in C[x] has a root in C.

30.3 Keywords

Normal Extension: 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].

Frobenius Automorphism: The Galois group of GF(p ) over GF(p) is cyclic of order n, generated
n
by the automorphism  defined by  (x) = x , for all x in GF(p ). This automorphism is usually
n
p
known as the Frobenius automorphism of GF(p ).
n
30.4 Review Questions
1. Compute each of the following Galois groups. Which of these field extensions are normal
field extensions? If the extension is not normal, find a normal extension of Q in which the
extension field is contained.
(a) G(Q( 30)/Q) (b) G(Q( 5)/Q)
4
(c) G(Q( 2, 3, 5)/Q) (d) G(Q( 2 , 2 ,i)/Q)
3
(e) G(Q( 6 ,i)/Q)
2. Let F K  E be field. If E is a normal extension of F, show that E must also be a normal
extension of K.
3. Let G be the Galois group of a polynomial of degree n. Prove that |G| divides n!.
4. Let F  E. If f(x) is solvable over F, show that f(x) is also solvable over E.

Answers: Self Assessment

1. (a) 2. (b) 3. (b) 4. (a) 5. (a)

30.5 Further Readings

Books Dan Saracino: Abstract Algebra; A First Course.
Mitchell and Mitchell: An Introduction to Abstract Algebra.
John B. Fraleigh: An Introduction to Abstract Algebra (Relevant Portion).

Online links www.jmilne.org/math/CourseNotes/
www.math.niu.edu
www.maths.tcd.ie/
archives.math.utk.edu

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