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

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

Question
Abstract algebra
asked 2020-11-11
An nth root of unity epsilon is an element such that \(\displaystyle\epsilon^{{n}}={1}\). We say that epsilon is primitive if every nth root of unity is \(\displaystyle\epsilon^{{k}}\) for some k. Show that there are primitive nth roots of unity \(\displaystyle\epsilon_{{n}}\in\mathbb{C}\) for all n, and find the degree of \(\displaystyle\mathbb{Q}\rightarrow\mathbb{Q}{\left(\epsilon_{{n}}\right)}\) for \(\displaystyle{1}\le{n}\le{6}\)

Answers (1)

2020-11-12
An nth root of unity ε is an element such that \(\displaystyleε{n}={1}\). It is said that ε is primitive if every nth root of unity is \(\displaystyleε{k}\) for some k. To show: There are primitive nth roots of unity:
\(\displaystyle\epsilon_{{n}}\in\mathbb{C}\) for all n
As it is given \(\displaystyleε\) is primitive and \(\displaystyleε{k}\) is the nth root of unity, by definition of nth root of unity we can say:
\(\displaystyle{\left(\epsilon^{{k}}\right)}^{{n}}={1}\)
Denote the nth root of unity \(\displaystyle\epsilon=\epsilon_{{k}}\) by the complex number:
\(\displaystyle{e}^{{\frac{{{2}\pi{i}}}{{k}}}}\), for some k
Obtain the value of \(\displaystyle\epsilon^{{k}}\)
\(\displaystyle\epsilon^{{k}}={\left(\epsilon^{{k}}\right)}^{{k}}={\left({e}^{{\frac{{{2}\pi{i}}}{{k}}}}^{k}={e}^{{{2}\pi{i}}}={1}{\left\langle.{e}^{{{2}\pi{i}}}={\cos{{2}}}\pi+{i}{\sin{{2}}}{i}={1}\right)}\right.}\)
As for some k becomes the primitive nth roots of unity. Hence it is proved that there are primitive nth roots of unity
\(\displaystyle\epsilon_{{n}}\in\mathbb{C}\) for all n
Find the degree of
\(\displaystyle\mathbb{Q}\rightarrow\mathbb{Q}{\left(\epsilon_{{n}}\right)}\) for \(\displaystyle{1}\le{n}\le{6}\)
Here, \(\displaystyle{Q}{\left(\epsilon_{{n}}\right)}\) is the field extension of the rational numbers generated over\(\displaystyle\mathbb{Q}\) by primitive th root of unity \(\displaystyle\epsilon_{{n}}\)
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