# Galoi group

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.

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HallsofIvy
Homework Helper
$x^5- 1= (x- 1)(x^4+ x^3+ x^2+ x+ 1)$ 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?

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
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