It is given that alpha is in F. Why does this imply that Z_p(alpha) is a subset of F?
2. What does it mean that “alpha is a generator of the multiplicative cyclic group F* of nonzero elements of F”? How does this show that Z_p(alpha)=F?
3. What is meant by |F|=p^n and how does this show that the degree of alpha over Z_p is n?

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It is given that alpha is in F. Why does this imply that Z_p(alpha) is a subset of F?

2. What does it mean that “alpha is a generator of the multiplicative cyclic group F* of nonzero elements of F”? How does this show that Z_p(alpha)=F?

3. What is meant by |F|=p^n and how does this show that the degree of alpha over Z_p is n?

11. Let

F

be a finite field of

p^{n}

elements containing the prime subfield

Z_{p}

. Show that if

? ? F

is a generator of the cyclic group

? F^{?}, ? ?

of nonzero elements of

F

, then

deg (?, Z_{p}) = n

.

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