How can I prove that F2 is a Field? I have to prove that the set {1,0} is a field. Do I have to go through every 8 possibilities when I want to show associativity or is there an easier way and how do you argue with an infinite set like $\mathbb{R}$ when you want to show that for ex. $(a + b) + c = a + (b + c)$, with $+$ being the normal addition.
I don’t know what you’re suppose to know or not.

It you write $\mathbb F_2= \mathbb Z /2 \mathbb Z$, then the result is clear. $\mathbb F_2$ is a field as it is the quotient of a ring over a maximal ideal and therefore is a field.

By the way, you’re almost forced to have this background. How do you define $\mathbb F_2$ without it?
Commutativity and associativity for both operations, as well as the distributive law all follow from the fact that these properties are satisfied in $\mathbb{R}$. After all, if you take $a,b,c\in\mathbb{F_2}$ then you can think of $a,b,c$ as of real numbers. Even in $\mathbb{R}$ we have $(a+b)+c=a+(b+c)$ so of course it is also true if you look at that number mod $2$. (if it is the same number then it also has the same remainder after division by $2$)