# About Relation between Cyclic Group and Finite Field

For prime $p$, the collection of nonzero element of a finite field $\mathbb{Z}_p$ is cyclic group with multiplication,namely $\mathbb{Z}_p^*$. And all cyclic group G with $n$ element is expressed as $\mathbb{Z}_n$ up to isomorphism.

Then, through both facts, Can I say that as follows?

• If $F$ is a finite field with $p^n$ element, then <$F^*, •$> is isomorphic to $\mathbb{Z}_{p^n-1}$.

If the answer is "yes", what is a isomorphism between <$\mathbb{Z}_p^*,•$> and< $\mathbb{Z}_{p-1},+$> when $n=1$ ??

• $F^*$ means all elements of the field $F$ except 0.
• • is a multiplication operation.

Regarding your second question, for $a$ a generating element of $F^*$, the isomorphism will be given by $a^n \mapsto n.1$.