EDIT: Hopefully question made clearer. Unfortunately this is a question found in analysis book and I do not actually have background on abstract algebra. Sorry for the confusion arisen.
As the title says, I am trying to show that a set of $\{0,1\}$, equipped with the obvious multiplication and with $1+1:=0$, is a field.
I encounter this question after the axioms of addition and multiplication of field in $\mathbb{R}$. I am not sure that how would I approach this question. Should I just check by brute force, e.g. check $x+y = y+x$ for all elements in $\{0,1\}$? How about associative of addition, i.e. $(x+y)+z= x+(y+z)$, how do I find myself such a third element $z$ from $\{0,1\}$?
Thanks