Suppose that F is a field of characteristic 0 and E is the splittingfield for some polynomial over F. If Gal(E/F) is isomorphic to D6,prove that there are exactly three fields L such that E ⊇ L ⊇ F and[E:L] = 6.

Suppose that F is a field of characteristic 0 and E is the splittingfield for some polynomial over F. If Gal(E/F) is isomorphic to D6,prove that there are exactly three fields L such that E ⊇ L ⊇ F and[E:L] = 6.

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

Suppose that F is a field of characteristic 0 and E is the splitting
field for some polynomial over F. If Gal(E/F) is isomorphic to D₆,
prove that there are exactly three fields L such that E ⊇ L ⊇ F and
[E:L] = 6.

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