(Solved): zn=1. It is called a primitive n-th root of unity if n is the smallest positi ...

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$$z^n=1.$$ It is called a primitive $$n$$-th root of unity if $$n$$ is the smallest positive integer for which the equation is satisfied. (a) (2 points) Justify that for each $$n$$ there are exactly $$n$$ different roots of unity (for example, for $$n=4$$ they are $$1,?1,\mathrm{i}$$ and $$?\mathrm{i}$$ ) (b) (2 points) Justify that for each $$n$$ there are exactly $$?(n)$$ different primitive roots of unity (for example, for $$n=4$$ they are $$i$$ and $$?i$$ ). (c) (1 point) Justify that the sum of all roots of unity is 0 . Hint: Factor the polynomial $$x^n?1$$.