# 9.7: Exercises

• 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.

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