Are the numbers 1, 0, -1 necessarily in every field? Do the numbers 1, 0, and -1 belong to every field? To me, this seems a fairly obvious conclusion of the field axioms, though I haven't seen it stated like so in any textbooks. 
 A: $0$ is in $F$, because $F$ is a group w.r.t. addition, it must have an identity element, we denote it as $0$.
$1$ is in $F$, because $F\backslash\{0\}$ is a group w.r.t. multiplication, it must have an identity element, we denote it as $1$.
$-1$ is in $F$, since $-1$ is the additive inverse of $1$ in the group $F$.
However, those elements can be the same, e.g. $-1=1$ in $\mathbb{Z}_2$.
A: The answer is yes and no.
Yes, every field has the elements $-1,0,1$. But, they are not necessarily the (real) numbers $-1,0,1$, they are just the elements given by the axioms. $1$ in the field doesn't mean the real number $1$.
Also, note that if your field has characteristic $2$, we have $-1=1$, so the three elements you listed are actually two.
Added: If the field has zero characteristic, then you can actually embed $\mathbb Z$ into the field. This, would allow us to view $-1, 0, 1$ as the integers $-1,0,1$, but we need to be very careful. What happens in this case, is that the field $k$ has a subring which is the same (i.e. isomorphic) as $\mathbb Z$. And note that in this case, even if we can identify a part of $k$ as "being" $\mathbb Z$, it is often the case that we cannot view all the elements in $k$ as "numbers". 
In finite characteristic you can only embed a factor of $\mathbb Z$ in the field, it would be wrong to think about $1$ as being a real number. The main issue is that in this case we have
$$1+1+1+....+1=0$$
A: All the integers are, although they might coincide.  Start with the multiplicative identity, and add it to copies of itself.  Or start with its additive inverse.
A: If you require $1,0$ and $-1$ to be unique, then no. For the Boolean numbers $\mathbb{F}_2$ contain only $(0,1)$ where $1$ is self inverse, ie $1+1=0$. But if $1$ is not self inverse (with respect to addition in your field) then the 3 numbers will be distinct. 
There's also the trivial example of the field with one element, but I don't think that's what your looking for.
