 # Prove if F(\sqrt[n]{a}) is unramified or totally ramified in certain hadaasyj 2022-04-28 Answered

Prove if $F\left(\sqrt[n]{a}\right)$ is unramified or totally ramified in certain conditions

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Step 1
For (1) ${x}^{n}-a$ is separable in the residue field $\frac{{O}_{F}}{\left({\pi }_{F}\right)}$ so $\frac{F\left({a}^{\frac{1}{n}}\right)}{F}$ is automatically unramified.
Note that Hensel lemma gives that ${\zeta }_{q-1}\in F\left({a}^{\frac{1}{n}}\right)$ where q is the cardinality of the residue field, and ${x}^{n}-a$ is separable in the residue field so Hensel lemma again gives that ${a}^{\frac{1}{n}}\in F\left({\zeta }_{q-1}\right)$ and $F\left({a}^{\frac{1}{n}}\right)=F\left({\zeta }_{q-1}\right)$.
For (2) take $nl+mv\left(a\right)=1$, let and ${x}^{n}-b$ is Eisenstein over ${O}_{F}$ so $\frac{F\left({b}^{\frac{1}{n}}\right)}{F}$ has degree n and $v\left({b}^{\frac{1}{n}}\right)=\frac{1}{n}$, it is totally ramified.