Let $\mathbb{K}$ be a field with a characteristic $P$, prove that if $P\neq{0}$, then $m$ is a prime number

Question

Let $\mathbb{K}$ be a field. Let's define a characteristic $P$ of $\Bbb{K}$ like this:

1. If $\sum_{i=1}^m 1\neq{0}$, then $P=0$
2. If $\sum_{i=1}^m 1={0}$ for some $m\geq{2}$ then $P$ is the lowest number $m$ with this property.

Now I'm asked to prove this two statements:

1. Show that if $P\neq{0}$, then $m$ is a prime number.
2. Show fields with characteristics equal to $0$, and others with chacteristics different from $0$.

but to be really honest, I'm not understanding this characteristic thing. So, right now it's kind of impossible for me to tackle the proofs if I'm not even understanding the definitions yet. I'm new to Linear Algebra but we are using some concepts of Abstract Algebra to enrich our classes.

I hope someone can explain me this and if you can, much appreciated! By the way, the book is in Portuguese (I don't know Portuguese), so the definitions/questions are a rough translation of what I grasped from the book.

I want to make clear that I'm not looking for a proof, I'm looking for an explanation so I can prove it myself.

Answer 1

First, the definition you're using is slightly too complex. The characteristic of a field is the smallest $p$ such that $\sum_{i=1}^p 1 = 0$. If that never happens, the field's characteristic is $0$.

Most fields that you're probably familiar with have characteristic $0$. I'll get you started with an example of a field that has characteristic $2$. Let $F=\{0, 1\}$ and define the field operations using arithmetic modulo $2$. Thus, for example, $1+1=0$ (because $2 \equiv 0 \pmod{2}$). I'll leave it to you to prove that $F$ defined with these operations really is a field.

Notice that if a field $F$ has characteristic $p$, then $\forall x \in F~\sum_{i=1}^p x=\sum_{i=1}^p 1 \cdot x = (\sum_{i=1}^p 1)x=0 \cdot x =0$.

This example may give you a hint to help you come up with other fields with non-zero characteristic. The key fact you need to remember about fields is that they have no zero divisors. That's how you prove that if a field has non-zero characteristic $p$, then $p$ must be prime.