Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 125 / 4 / b

Assume ${\mu }_n\subset K$ and put $\alpha =\sqrt[n]{a}$. All roots of $X^n-a$ are $\zeta \alpha$ with $\zeta \in {\mu }_n\subset K$, so $K(\alpha )$ is its splitting field and is a Finite Galois extension. The mapis an injective group homomorphism.