An nth root of unity epsilon is an element such that ϵn=1. We say that epsilon is primitive if every nth root of unity is ϵk for some k. Show that there are primitive nth roots of unity ϵn∈C for all n, and find the degree of Q→Q(ϵn) for 1≤n≤6

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Laith Petty · Accepted Answer

An nth root of unity ε is an element such that εn=1. It is said that ε is primitive if every nth root of unity is εk for some k. To show: There are primitive nth roots of unity: ϵn∈C for all n As it is given ε is primitive and εk is the nth root of unity, by definition of nth root of unity we can say: (ϵk)n=1 Denote the nth root of unity ϵ=ϵk by the complex number: e2πik, for some k Obtain the value of ϵk ϵk=(ϵk)k=(e((2π)/k)k=e(2π)=1(:.e(2π)=cos⁡2π+isin⁡2i=1) As for some k becomes the primitive nth roots of unity. Hence it is proved that there are primitive nth roots of unity ϵn∈C for all n Find the degree of Q→Q(ϵn) for 1≤n≤6 Here, Q(ϵn) is the field extension of the rational numbers generated overQ by primitive th root of unity ϵn

An nth root of unity epsilon is an element such that epsilon^n=1. We say that epsilon is primitive if every nth root of unity is epsilon^k for some k. Show that there are primitive nth roots of unity epsilon_n in CC for all n, and find the degree of QQ rarr QQ(epsilon_n) for 1<=n<=6

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