The roots of this polynomial are $(-1)\zeta^n$. Where $\zeta$ is the 5th root of unity. So the splitting field should just be $\mathbb{Q}(\zeta)$. This has galois group $\mathbb{Z}/4$ since it's just a cyclotomic extension. Is this correct?
Source: https://dornsife.usc.edu/assets/sites/363/docs/F17_510ab.pdf