I have the following field theory question: I am given this polynomial $ x^5-5 $ for which I am supposed to find a basis for the splitting field over Q all I can determine in this regard is that it has order 20 I was never really taught how to find something like a basis for a cyclotomic field all I know is that the splitting field is $ Q(\sqrt{5},\zeta_5) $ Could you please help me find a Q basis and explain to me how to do it please? Thank you
2026-03-26 21:07:19.1774559239
Q basis for splitting field
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You can use what is called a “primitive element” of K for what you can take an element x such that the images of x by the 20 Q-automorfisms of K (splitting field) be distinct. You can verify that the simple sum of your elements satisfy this condition.