9.7: Exercises

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Bob Dumas and John E. McCarthy
University of Washington and Washington University in St. Louis

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EXERCISE 9.1. What are the primitive fourth roots of unity?

EXERCISE 9.2. Show that if \(\omega\) is any \(n^{\text {th }}\) root of unity other than 1, then \(1+\omega+\omega^{2}+\cdots+\omega^{n-1}=0 .\) EXERCISE 9.3. How many primitive cube roots of unity are there? How many primitive sixth roots? How many primitive \(n^{\text {th }}\) roots for a general \(n\) ?

EXERCISE 9.4. Redo Example \(9.8\) to get all three roots from the Tartaglia-Cardano formula.

EXERCISE 9.5. Let \(p(x)=x^{3}+3 x+\sqrt{2}\). Show without using the Cardano-Tartaglia formula that \(p\) has exactly one real root. Find it. What are the complex roots?

EXERCISE 9.6. Fill in the proof of Proposition 9.34.

EXERCISE 9.7. Let \(g: G \rightarrow \mathbb{C}\) be a continuous function on \(G \subseteq \mathbb{C}\). Show that \(\Re(g), \Im(g)\) and \(|g|\) are continuous. Conversely, show that the continuity of \(\Re(g)\) and \(\Im(g)\) imply the continuity of \(g\).

EXERCISE 9.8. Show that every continuous real-valued function on a closed, bounded subset of \(\mathbb{C}\) attains its extrema.