You have seen answers for the first question. For the second question:
The answer won't be unique. To construct a field of order $p^n$ for a prime $p$ you consider the polynomial ring $F_p[x]$ and a monic ireducible polynomial $f$ of degree $n$. Here $p=2$ and $n=3$.
Note that a polynomial of degree $3$ is reducible if and only if it has a zero. Hence we want a monic polynomial of degree $3$ in $F_2[x]$ with no zeros.
It will be of the form
where $a,b,c\in F_2$. Now $0$ shan't be a zero, so $c=1$. Then $1$ shan't be a zero as well so $(a,b)=(1,0)$ or $(a,b)=(0,1)$.
We end up with two options
$$f(x)=x^3+x^2+1$$ or $$g(x)=x^3+x+1$$.
In $F_2[x]/f$ we have $x^4=x\cdot x^3=x(x^2+1)=x^3+x=x^2+x+1$.
In $F_2[x]/g$ we have $x^4=x\cdot x^3=x(x+1)=x^2+x$.
You see that the answer is not unique.