Find the Galois group of the polynomial $f(x)=x^{4}+x^{3}+1$ over (i) the finite field $\mathbf{F}_{2}$, (ii) the finite field $\mathbf{F}_{3}$, (iii) the finite field $\mathbf{F}_{4}$, (iv) the field $\mathbf{Q}$ of rational numbers.

[Results from the course which you use should be stated precisely.]