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P. 280
Unit 29: Computing Galois Groups
29.5 Review Questions Notes
1. Give the order and describe a generator of the Galois group of GF (729) over GF(9).
2. Determine the Galois group of each of the following polynomials in Q[x]; hence, determine
the solvability of each of the polynomials
(a) x 12x + 2 (b) x 4x + 2x + 2
2
5
5
4
(c) x 5 (d) x x 6
2
3
4
(e) x + 1 (f) (x 2) (x + 2)
2
5
2
(g) x 1 (h) x + 1
8
8
(i) x 3x 10
2
4
3. Find a primitive element in the splitting field of each of the following polynomials in
Q[x].
(a) x 1 (b) x 2x 15
2
4
4
(c) x 8x + 15 (d) x 2
3
2
4
4. Prove that the Galois group of an irreducible quadratic polynomial is isomorphic to Z .
2
5. Prove that the Galois group of an irreducible cubic polynomial is isomorphic to S or Z .
3 3
Answers: Self Assessment
1. (a) 2. (b) 3. (a) 4. (a)
29.6 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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