To every finite field K there exists a positive integer m sucht that m*a = a + a + ... + a ( a times ) = 0 for every a in K

Question

How can i prove this:

1. To every finite field K there exists a positive integer m sucht that m*a = a + a + ... + a ( a times ) = 0 for every a in K.

2. The smallest positive integer m > 0 in 1. is a prime number.

Answer 1

Note that in a finite field $(F,+,.)$ ,$(F,+)$ is a finite group with zero element $0_F$.

In a finite group every element has finite order .Hence for each $a\in F$ there exists $n\in \Bbb N$ such that $n.a=0$

Answer 2

Every field contains $1$ and $0$ by definition. The characteristic of a finite field must be a prime number, see here. Then $1+1\cdots +1=0, i.e., p\cdot 1=0$. Hence $p\cdot a=0$.