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(d) Define the semidirect product Z/5Z × (Z/5Z)* to have Z/5Z × (Z/5Z)* as its underlying set, but with binary operation (a, b) · (c,d) = (a + bc, bd). . Show that Z/5Z × (Z/5Z)* is a group with this operation and that it is nonabelian. (e) Show that o: G(K/Q) → Z/5Z × (Z/5Z)* is a group isomorphism.

(d) Define the semidirect product Z/5Z × (Z/5Z)* to have Z/5Z × (Z/5Z)* as its underlying set, but with binary operation (a, b) · (c,d) = (a + bc, bd). . Show that Z/5Z × (Z/5Z)* is a group with this operation and that it is nonabelian. (e) Show that o: G(K/Q) → Z/5Z × (Z/5Z)* is a group isomorphism.

Algebra & Trigonometry with Analytic Geometry
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter3: Functions And Graphs
Section3.3: Lines
Problem 27E
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(d) Define the semidirect product Z/5Z × (Z/5Z)* to have Z/5Z × (Z/5Z)* as its underlying set, but with binary operation (a, b) (c,d) = (a + bc, bd). Show that Z/5Z × (Z/5Z)* is a group with this operation and that it is nonabelian. (e) Show that o: G(K/Q) → Z/5Z × (Z/5Z)* is a group isomorphism. .
Transcribed Image Text:(d) Define the semidirect product Z/5Z × (Z/5Z)* to have Z/5Z × (Z/5Z)* as its underlying set, but with binary operation (a, b) (c,d) = (a + bc, bd). Show that Z/5Z × (Z/5Z)* is a group with this operation and that it is nonabelian. (e) Show that o: G(K/Q) → Z/5Z × (Z/5Z)* is a group isomorphism. .
Exercise 1: The point of this exercise is to calculate G(K/Q) where K is the splitting field of x5 - 2. We'll take it in steps. Throughout, let K be the splitting field of x² - 5 and let C = e²ri/5 be a primitive 5th root of unity.
Transcribed Image Text:Exercise 1: The point of this exercise is to calculate G(K/Q) where K is the splitting field of x5 - 2. We'll take it in steps. Throughout, let K be the splitting field of x² - 5 and let C = e²ri/5 be a primitive 5th root of unity.
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