Page 301 - DMTH403_ABSTRACT_ALGEBRA

Abstract Algebra

Notes (b) Show that G((w)/) is abelian of order p – 1.
(c) Show that the fixed field of G(()/) is .

2. Let F be a finite field of characteristic zero. Let E be a finite normal extension of F with
Galois group G(E/F): Prove that F  K  L  E if and only if {id}  G(E/L)  G(E/K) 
G(E/F).
3. Let F be a field of characteristic zero and let f(x)  F[x] be a separable polynomial of degree
n. If E is the splitting field of f(x), let  ,..., be the roots of f(x) in E. Let    i j (   j ).
i
1
n

We define the discriminant of f(x) to be  . 2
(a) If f(x) = ax + bx + c, show that  = b – 4ac.
2
2
2
(b) If f(x) = x + px + q, show that  = –4p – 27q .
2
2
3
3
(c) Prove that  is in F.
2
(d) If   G(E/F) is a transposition of two roots of f(x), show that () = –.
(e) If   G(E/F) is an even permutation of the roots of f(x), show that () = .
(f) Prove that G(E/F) is isomorphic to a subgroup of A if and only if   F.
n
(g) Determine the Galois groups of x + 2x – 4 and x + x – 3.
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3
Answers: Self Assessment
1. (b) 2. (a) 3. (b) 4. (b)
32.6 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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