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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.
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The answer to you first question is yes. The non-zero elements of a finite field are a cyclic group for multiplication.

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

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