Constructing a multiplication table for a finite field 
Let $f(x)=x^3+x+1\in\mathbb{Z}_2[x]$ and let $F=\mathbb{Z}_2(\alpha)$, where $\alpha$ is a root of $f(x)$. Show that $F$ is a field and construct a multiplication table for $F$.

Can you please help me approach this problem? I've tried searching around, but I don't really know what I'm looking for!
Thanks.

By the division algorithm, any polynomial $g\in\mathbb{Z}_2[x]$ can be uniquely written as
$$g=a_0+a_1x+a_2x^2+qf$$
for some $q\in\mathbb{Z}_2[x]$ and some $a_0,a_1,a_2\in\mathbb{Z}_2$ (depending on $g$, of course). Thus, the quotient ring $\mathbb{Z}_2[x]/(f)$ consists precisely these eight cosets (corresponding to each possible choice of the $a_i$):
$$\begin{array}{cc}
0 + (f) &\quad 1 + (f) \\
x + (f) &\quad 1 + x + (f) \\
x^2 + (f) &\quad 1 + x^2 + (f) \\
x + x^2 + (f) &\quad 1 + x + x^2 + (f) \\
\end{array}$$
Use the definition of addition and multiplication in a quotient ring to construct the multiplication table. For example,
$$\begin{align*}
\biggl[x + (f)\biggr]\cdot \biggl[x^2 + (f)\biggr]&=x^3 + (f)\\\\
&= \biggl[0 +(f)\biggr] + \biggl[x^3+(f)\biggr]\\\\
&= \biggl[f +(f)\biggr] + \biggl[x^3+(f)\biggr]\\\\
&= \biggl[1 + x + x^3+(f)\biggr] + \biggl[x^3+(f)\biggr]\\\\
&= 1 + x + 2x^3+(f)\\\\
&=1+x+0x^3+(f)\\\\
&=1+x+(f)
\end{align*}$$
You can prove that $F\cong\mathbb{Z}_2[\alpha]\cong\mathbb{Z}_2[x]/(f)$ is a field because: $\mathbb{Z}_2[x]$ is a PID, hence a non-zero ideal of $\mathbb{Z}_2[x]$ is maximal iff it is prime iff it is generated by an irreducible element, so $\mathbb{Z}_2[x]/(f)$ is a field iff $f$ is irreducible, and you can either check directly that $f$ doesn't factor non-trivially, or observe that since $\deg(f)\leq 3$ it suffices to check that $f$ has no roots in $\mathbb{Z}_2$, which it doesn't because $f(0)=1$ and $f(1)=1$.
