Structure of the multiplication operation in finite fields.

Question

The multiplication group in finite fields is isomorphic to $Z_{p^n-1}$. But isnt the order of a multiplicative group mod $p^n$ , $p^n-p^{n-1}$. Is the multiplication group in finite fields just a matter of definition then? And how do you prove that there are no other finite fields apart from the ones of this order?

Answer 1

You are confusing multipicative and additive structure. The multiplicative group of a finite field is isomorphic to the additive group $\mathbb Z/(p^{n}-1)\mathbb Z$, simply because both groups are cyclic of the same order. To see that every finite field has order $p^n$, note that if $p,q$ are distinct primes dividing the order of a finite field $K$, then there is an element $a$ of order $p$ and an element $b$ of order $q$. Thus, $qa\neq 0$ and $pb\neq 0$ but $(qa)(pb)=0$, contradiction because a field has no zero divisors.