6. Let K be the field as in Problem 5. Show that K is the splitting field over Q, and, therefore, the isomorphisms of K (onto subfields of Q leaving Q fixed) determined in Problem 5 are precisely the elements of G(K/Q). Moreover, show that K is separable over Q. Also find X(Q(/2)) where A as in the Main Theorem of Galois Theory.
6. Let K be the field as in Problem 5. Show that K is the splitting field over Q, and, therefore, the isomorphisms of K (onto subfields of Q leaving Q fixed) determined in Problem 5 are precisely the elements of G(K/Q). Moreover, show that K is separable over Q. Also find X(Q(/2)) where A as in the Main Theorem of Galois Theory.
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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Number 6: show at K is splitting over Q.

Transcribed Image Text:6. Let K be the field as in Problem 5. Show that K is the splitting field over Q, and, therefore, the
isomorphisms of K (onto subfields of Q leaving Q fixed) determined in Problem 5 are precisely the
elements of G(K/Q). Moreover, show that K is separable over Q. Also find X(Q(v2)) where A as
in the Main Theorem of Galois Theory.
![5. Let K = Q(V2, i). Find all isomorphisms of K onto a subfield Q as extensions of the identity
automorphism o :Q → Q. [Hint: Use the tower Q < Q(/2) < Q(v2)(i).]](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fb4dd8e23-ab66-4b24-8e54-a64daec9031c%2Ffcadf9b7-f916-4eef-9db0-6fdf2d7387b5%2F0krnsbp_processed.jpeg&w=3840&q=75)
Transcribed Image Text:5. Let K = Q(V2, i). Find all isomorphisms of K onto a subfield Q as extensions of the identity
automorphism o :Q → Q. [Hint: Use the tower Q < Q(/2) < Q(v2)(i).]
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