I was reviewing the definition of the characteristic of a field where it was defined to be:
"The characteristic of a field $F$, denoted ch$(F)$, is defined to be the smallest positive integer $p$ such that $p\cdot 1_F = 0$ if such a p exists and is defined to be $0$ otherwise."
The further proposition said:" If ch$(F) = p$ then for any $\alpha \in F. p\cdot \alpha=\alpha+\alpha+...(p\, times)+\alpha=0$.
My question was that:
how was the addition associated multiplication?
Further, how did the prime elements associated with the prime number?
Where I found a sentence at the beginning of the chapter: "For $n$ a positive integer, let $n\cdot 1_F = 1_F+...+ 1_F$ ($n$ times). "
- if we didn't have this relation, does that mean the above proposition would be incorrect and we can no longer associate the addition with multiplication?