Characteristic of a finite field - equivalent formula

Question

Let $\mathbb{K}$ be a field with characteristic $\text{char}(\mathbb{K}) = p$ where $p$ is a prime. It means that the sum of $p$ unities ($1+ 1+\ldots + 1$ added $p$ times) equals $0$.

How to show that it is also correct for an arbitrary element $a\in\mathbb{K}?$ I mean taking $a + a+\ldots +a =0$ if added $p$ times...

Answer 1

It follows immediately from the distributivity of multiplication over addition: $$a+a+\ldots+a=a\cdot(1+1+\ldots+1)=a\cdot0=0.$$

Answer 2

Hint

$$a+a+a+...+a=a(1+1+1+...+1)$$