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 = e2i/5 be a primitive 5th root of unity. (a) Calculate [K: Q], hence #G(K/Q). (Hint: Show that [K: Q] must be divisible by both 4 and 5.) (b) Similarly to what we did in class, define two functions, λ: G(K/Q) → Z/5Z and k: G(K/Q) → (Z/5Z)* by writing o(√√/2) = ¢¹(0) √√/2_and_σ($)=¢*(º). Show that the function 6: G(K/Q) → Z/5Z × (Z/5Z)*, (0) = (λ(0), k(0)) is a bijection of sets. (Hint: First show it is injective, then compare the number of elements.) (c) Show that X(OT) = X(0) + K(0)X(T) K(OT) = K(0)K(T). and (d) Define the semidirect product Z/5Z × (Z/5Z)* to have Z/5Z × (Z/5Z)* as its underlying set, but with binary operation (a, b) · (c,d) = (a + bc, bd). Show that Z/5Z × (Z/5Z)* is a group with this operation and that it is nonabelian. (e) Show that : G(K/Q) → Z/5Z × (Z/5Z)* is a group isomorphism.
Exercise 1: The point of this exercise is to calculate G(K/Q) where K is the splitting field of x5 - 2. We'll take it in steps. Throughout, let K be the splitting field of x² - 5 and let = e2i/5 be a primitive 5th root of unity. (a) Calculate [K: Q], hence #G(K/Q). (Hint: Show that [K: Q] must be divisible by both 4 and 5.) (b) Similarly to what we did in class, define two functions, λ: G(K/Q) → Z/5Z and k: G(K/Q) → (Z/5Z)* by writing o(√√/2) = ¢¹(0) √√/2_and_σ($)=¢*(º). Show that the function 6: G(K/Q) → Z/5Z × (Z/5Z)*, (0) = (λ(0), k(0)) is a bijection of sets. (Hint: First show it is injective, then compare the number of elements.) (c) Show that X(OT) = X(0) + K(0)X(T) K(OT) = K(0)K(T). and (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 : G(K/Q) → Z/5Z × (Z/5Z)* is a group isomorphism.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![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 = e2mi/5 be a primitive 5th root of unity.
(a) Calculate [K: Q], hence #G(K/Q). (Hint: Show that [K: Q] must be
divisible by both 4 and 5.)
(b) Similarly to what we did in class, define two functions, λ: G(K/Q) → Z/5Z
and ê: G(K/Q) → (Z/5Z)* by writing
o(√√2) = (0)√√2 and o($) = (o).
Show that the function
¢: G(K/Q) → Z/5Z × (Z/5Z)*,
þ(0) = (\(0), k(0))
is a bijection of sets. (Hint: First show it is injective, then compare the
number of elements.)
(c) Show that
X(OT) = X(0) + K(0)】(T)
K(OT) = K(0)K(T).
and
(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 : G(K/Q) → Z/5Z × (Z/5Z)* is a group isomorphism.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F081e66de-98dd-47c4-ad89-12bf6ce96be8%2Fc8553ebc-edca-4e98-9cd1-3ab149139140%2Fagc5gd7_processed.png&w=3840&q=75)
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 = e2mi/5 be a primitive 5th root of unity.
(a) Calculate [K: Q], hence #G(K/Q). (Hint: Show that [K: Q] must be
divisible by both 4 and 5.)
(b) Similarly to what we did in class, define two functions, λ: G(K/Q) → Z/5Z
and ê: G(K/Q) → (Z/5Z)* by writing
o(√√2) = (0)√√2 and o($) = (o).
Show that the function
¢: G(K/Q) → Z/5Z × (Z/5Z)*,
þ(0) = (\(0), k(0))
is a bijection of sets. (Hint: First show it is injective, then compare the
number of elements.)
(c) Show that
X(OT) = X(0) + K(0)】(T)
K(OT) = K(0)K(T).
and
(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 : G(K/Q) → Z/5Z × (Z/5Z)* is a group isomorphism.
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