Galoi group

  • Thread starter algekkk
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  • #1
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Anybody can help me show that Gal(E/Q) is isomorphic to Z4? E is the splitting field for X^5-1 over Q. Thanks.
 

Answers and Replies

  • #2
HallsofIvy
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[itex]x^5- 1= (x- 1)(x^4+ x^3+ x^2+ x+ 1)[/itex] has the single real root, x= 1, and 4 complex roots, [itex]e^{2\pi i/5}[/itex], [itex]e^{4\pi i/5}[/itex], [itex]e^{6\pi i/5}[/itex], and [itex]e^{8\pi i/5}[/itex]. Can you construct the Galois group from that? What does Z4 look like?
 
  • #3
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Z4 is {0,1,2,3} I can tell that their orders are all four. Just not sure about what's the rest needed to show isomorphic.
 
  • #4
Office_Shredder
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What orders are all four? Only two elements of Z4 have an order of 4.

Do you know what a relationship between the non-trivial roots of x5-1 is that allows you to describe all the roots in terms of one of them?
 
  • #5
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Ok, thanks for the help. I have this one solved. All I need to do is to show the Galois group are cyclic.
 

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