Let E be the splitting field of x4 + 1 over Q. Find Gal(E/Q). Find allsubfields of E. Find the automorphisms of E that have fixed fieldsQ(√2), Q(√-2 ), and Q(i). Is there an automorphism of E whosefixed field is Q?

Let E be the splitting field of x4 + 1 over Q. Find Gal(E/Q). Find allsubfields of E. Find the automorphisms of E that have fixed fieldsQ(√2), Q(√-2 ), and Q(i). Is there an automorphism of E whosefixed field is Q?

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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Question

Let E be the splitting field of x⁴ + 1 over Q. Find Gal(E/Q). Find all
subfields of E. Find the automorphisms of E that have fixed fields
Q(√2), Q(√-2 ), and Q(i). Is there an automorphism of E whose
fixed field is Q?

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