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 map
$$
\operatorname{Gal}(K(\alpha)/K)\longrightarrow\mu_n,
\qquad
\sigma\longmapsto\frac{\sigma(\alpha)}{\alpha}
$$
is an injective group homomorphism.

Solution

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

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-125/4/b/solution