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# 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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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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