Characteristic of Finite Fields

Question

Must all finite fields have a non-zero characteristic? If so how would I go about proving this? If this is true it would seem to follow that every member of the field can be represented as being equal to a sum of n 1's where n < the characteristic. Would I be correct in assuming this?

Answer 1

Yes, every finite field has positive characteristic, because any field of characteristic zero contains a copy of $\mathbb{Z}$, hence is infinite.

However, if a finite field $F$ has $p^d$ elements with $d>1$, then not every element will have the form $n\cdot 1$, since there are only $p$ such elements.

Answer 2

If $\mathrm{char}R=0$ this means that the sum $n \cdot 1$ is never $0$. This means that all such sums are different, so $R$ necessarily has infinitely many elements.

Also, be a little careful, because there do exist infinite fields of prime characteristic.