Week 8 Exercises, due 16 September 2021

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Exercises 7.11 Week 8 Exercises, due 16 September 2021

Let \(n\in\NN\text{.}\) An \(n\)th root of unity is a complex number \(z\in\CC\) such that \(z^n = 1\text{.}\) We have seen that the set of roots of unity is the set

\begin{equation*} \mu_n = \{1, \zeta_n, \zeta_n^2, \ldots, \zeta_n^{n-1}\}\text{,} \end{equation*}

where \(\zeta_n = \exp(i2\pi/n)\text{.}\) An \(n\)-th root of unity \(z\in\mu_n\) is said to be primitive if and only if, for every \(w\in\mu_n\text{,}\) there exists \(k\in\NN_0\) such that \(w = z^k\text{.}\)

1.

Let \(n\in\NN\text{.}\) Prove that the roots of unity sum to \(0\text{;}\) that is:

\begin{equation*} \sum_{z\in \mu_n} z = 0\text{.} \end{equation*}

2.

How many primitive \(n\)-th roots of unity are there when \(n=4\text{?}\) \(n=5\text{?}\) \(n=6\text{?}\)

3.

Identify all the roots of all the following polynomials: \(z-1\text{,}\) \(z+1\text{,}\) \(z^2+z+1\text{,}\) \(z^2+1\text{,}\) \(z^2-z+1\text{,}\) \(z^4-z^2+1\text{.}\)

4.

Find all the solutions to the equation \(z^2-\overline{\zeta}_3z+\zeta_3=0\text{.}\)