I'm looking for the minimal polynomial of $\alpha = i\frac{\sqrt{3}}{2}+\frac{1}{2}$ in $\mathbb{Q}[x]$. A polynomial with a root $\alpha$ is
$(2(x-\frac{1}{2}))^4-9.$
A computer algebra system shows me that the polynomial is irreducible. I looking for a way to compute this by hand in an exam. Is there any way to do so without using any irreduciblity criterions which where not part of my lecture?