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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 # 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 $$\displaystyle\epsilon_{{n}}$$

### Relevant Questions asked 2021-05-05

A random sample of $$n_1 = 14$$ winter days in Denver gave a sample mean pollution index $$x_1 = 43$$.
Previous studies show that $$\sigma_1 = 19$$.
For Englewood (a suburb of Denver), a random sample of $$n_2 = 12$$ winter days gave a sample mean pollution index of $$x_2 = 37$$.
Previous studies show that $$\sigma_2 = 13$$.
Assume the pollution index is normally distributed in both Englewood and Denver.
(a) State the null and alternate hypotheses.
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(b) What sampling distribution will you use? What assumptions are you making? NKS The Student's t. We assume that both population distributions are approximately normal with known standard deviations.
The standard normal. We assume that both population distributions are approximately normal with unknown standard deviations.
The standard normal. We assume that both population distributions are approximately normal with known standard deviations.
The Student's t. We assume that both population distributions are approximately normal with unknown standard deviations.
(c) What is the value of the sample test statistic? Compute the corresponding z or t value as appropriate.
(Test the difference $$\mu_1 - \mu_2$$. Round your answer to two decimal places.) NKS (d) Find (or estimate) the P-value. (Round your answer to four decimal places.)
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At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are not statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we fail to reject the null hypothesis and conclude the data are statistically significant.
At the $$\alpha = 0.01$$ level, we reject the null hypothesis and conclude the data are not statistically significant.
(f) Interpret your conclusion in the context of the application.
Reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Reject the null hypothesis, there is sufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
Fail to reject the null hypothesis, there is insufficient evidence that there is a difference in mean pollution index for Englewood and Denver.
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(Round your answers to two decimal places.)
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upper limit
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Because the interval contains only positive numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, we can not say that the mean population pollution index for Englewood is different than that of Denver.
Because the interval contains both positive and negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is greater than that of Denver.
Because the interval contains only negative numbers, this indicates that at the 99% confidence level, the mean population pollution index for Englewood is less than that of Denver. asked 2021-03-02

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