Is subtraction defined in $\mathbb{Z}_{2}$? I know $(\mathbb{Z}_{2},+,*)$ is a field but I'm wondering if substraction is defined in it. I ask this because if I want to create a new Field $F:=\mathbb{Z}_{2}\times \mathbb{Z}_{2}$ with the "Addition": $(a,b)\bigoplus(c,d)=(a-b,c-d)$ is this possible?

Adding in details: In $\mathbb Z_2$, every element is its own additive inverse: $0+0=0$, $1+1=0$, so $0=-0$, $1=-1$. Another way of saying this is that $1+1=0$, and it makes this a "Field of Characteristic 2".

So, $\forall b\in \mathbb Z_2,b=-b$.

Now, subtraction is defined as adding the additive inverse in general, so the way we define subtraction is $a-b=a+(-b)$. But in fields of characteristic 2, such as $\mathbb Z_2,$ $-b=b$, so we can simplify this to $a-b=a+(-b)=a+b$