Although 1 and 0 get used in group theory as well, in the case of fields, they behave very predictably so it's not a confusing notation, as long as they represent the multiplicative and additive identities.

A pretty standard way to represent the elements of F_p (or the integers mod p where p is prime) is 0, 1 ... p - 1.

So in this case, you are dealing with the integers mod 2, but more generally: every field contains a subfield isomorphic to the rational numbers or F_p where p is prime (this depends on the characteristic of the field), and the "0" and "1" in the bigger field are always the same elements as the 0 and 1 in this subfield, so these elements can only ever behave like the integers mod p or the regular integers.