Showing Gal(E/Q) is Isomorphic to Z4

[algekkk]

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.

 

[HallsofIvy]

$x^5- 1= (x- 1)(x^4+ x^3+ x^2+ x+ 1)$ has the single real root, x= 1, and 4 complex roots, $e^{2\pi i/5}$, $e^{4\pi i/5}$, $e^{6\pi i/5}$, and $e^{8\pi i/5}$. Can you construct the Galois group from that? What does Z4 look like?

 

[algekkk]

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.

 

[Office_Shredder]

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?

 

[algekkk]

Ok, thanks for the help. I have this one solved. All I need to do is to show the Galois group are cyclic.