in my lecture about field extension and tower theorem for extension degree, the professor gives the following example :
$$\mathbb{Z}_2 \le \frac{\mathbb{Z}_2[x]}{(x^2 + x + 1)} \le \frac{(\frac{\mathbb{Z}_2[x]}{(x^2 + x + 1)})[y]}{(y^3 + y + 1)} $$ has degree $3 * 2 = 6$, and a basis consists of $$1, \bar{x},\bar{y}, \bar{xy}, \bar{y^2}, \overline{xy^2}$$
This finite field thus has $2^6 $ elements.
It is the very first time he uses the notation $\bar{x}$ with this horizontal bar above $x$ and I can't understand what it means. Is it an arbitrary root of the poly $x^2 + x + 1$ ? Can someone explain ? Thanks!