In communication theory classes I recall this sort of extension of $\mathbb{Z}/2\mathbb{Z}$ where an imaginary $\alpha$ is defined so that $\alpha^2+\alpha+1 = 0$. Then more such imaginaries must be added so that other polynomials have solutions.
Can anyone remind me what this set is called?
On
The name you are looking is a "field extension" that is generated by the root of the polynomial $x^2 + x + 1$ ; another notation for this is $GF(4)$ due to the fact that there exists only one field up to isomorphism of degree $p^n$ when $n \ge 1$ and $p$ is a prime, which we denote $GF(p^n)$ (here we're looking at $2^2 = 4$, so $p=2$ is prime and $n=2$). This is a result from Galois Theory, if you're interested in reading more about it.
Hope that helps,
EDIT : After noticing what Michael Hardy said, I didn't read the question completely. If you want to continue "adding roots" to allow other polynomials to "split" too (i.e. have their roots in the field), then that is called the "algebraic closure" of your field, i.e. the field that contains $\mathbb Z / 2 \mathbb Z$ as a subfield and contains the roots of every polynomial with coefficients in itself.
The "algebraic closure" of the field $\mathbb{Z}_2$ is the smallest field containing $\mathbb{Z}_2$ in which all polynomials with coefficients in the field have zeroes in the field.
There's a simple proof that the algebraic closure must be infinite: if you have only finitely many members $a$, then $$f(x) = 1+\prod_{a} (x-a)$$ is a polynomial with no roots within the field.