Char F = 2 basic understanding

Question

I've started learning Linear Algebra and am having trouble properly understanding the $char \mathbb F = 2$

The question I need to solve is:

Let $\mathbb F$ be a field and $a,b \in \mathbb F $ prove that:

1. $a + b = a − b$
2. $(a + b)^2 = a^2 + b^2$

Now, I don't want solutions to these two question but a better understatement of what the $char(\mathbb F) = 2$ actually means. From searching for similar questions I saw it is related to ring theory and from what I understand it is basically like grouping numbers by their $[X]\mod2$

|  0  |  1  |
+-----------+
|  0  |  1  |
|  2  |  3  |
|  4  |  5  |
+-----------+  //and so on  

But I can't get a proper understanding if it is really like this.

Answer 1

The characteristic is simply the (minimal) number of times you need to add $1$ to itself to get back to $0$; and we say a field has characteristic $0$ if it has no finite characteristic by that previous definition. For example, $\mathbb{Q}$ has characteristic $0$; the field $\mathbb{F}_9$ has characteristic $3$; the field $\mathbb{F}_{17}$ has characteristic $17$.

In a field of characteristic $2$, it is the case that $1+1 = 0$. Therefore, for example, $a+a = a(1+1) = a \times 0 = 0$.

You might be thinking too much about fields which are "basically $\mathbb{Q}$" or "basically $\mathbb{R}$". When thinking about fields whose characteristic is not $0$, it's often helpful to abandon your intuition about how fields behave, and just think of them as sets with certain useful operations that mean elementary algebra works on them.

Answer 2

The characteristic of a field $F$

is the smallest positive integer $n$ such that $n.a=0_F\forall a\in F$.

is the smallest positive integer $n$ such that $n.1_F=0_F$.

The above two statements are equivalent and it will be a good exercise for you to prove their equivalence.

If no such integer $n$ exists then characteristic of a field $F$ is defined to be $0$