Exercises: complex numbers and Fⁿ — Studies in Algebra and Group Theory

N15 Exercises: complex numbers and Fⁿ

Exercise N15.1. Show that for all .

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and .

Exercise N15.2. Show that for all .

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Exercise N15.3. Show that for all .

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Exercise N15.4. Show that for all .

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Exercise N15.5. For all there is such that .

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Consider any ; notice , so .

Exercise N15.6. Show that for all with there is a unique such that .

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Exercise N15.7. Show that is a cube root of .

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so the cube is .

Exercise N15.8. Find the square roots of .

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. Since , Roots: and .

Exercise N15.9. Find such that .

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Exercise N15.10. Explain why there is no such that .

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would have to equal each of The first two both equal , but

Exercise N15.11. Show that for all .

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, , . Each , , is an element of , and is either or , in either of which .

Exercise N15.12. Show that for all and all .

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Exercise N15.13. Show that for all .

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Exercise N15.14. Show that for all and all .

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Exercise N15.15. Show that for all and all .

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