In prop 7.11 of Neukirch's Algebraic Number Theory says The maximal tamely ramified subextension $V/K$ of $L/K$ has value group $w(V^*)=w(L^*)^{(p)}$, where $L/K$ is an algebraic extension of henselian field, and $w(L^*)^{(p)}/v(K^*)$ consists of all elements of $w(L^*)/v(K^*)$ whose order is prime to $p$.
In its proof, it assumes that $K$ is totally ramfied, and says: as in the proof of (7.7), for every $\omega \in w(L^*)^{(p)}$, we can find a radical of $K$ of the same value.
In the proof of (7.7), Neukirch shows that if $L/K$ is totally, tamely ramified, then $L$ is generated by radicals of $K$ as follows:
for any representative $w$ for the quotient value groups $w(L^*)/v(K^*)$, we could find an element $\alpha \in L^*$ s.t. $w(\alpha)=w$, and $w(\alpha^m) \in v(K^*)$ for some $m$ prime to $p$, $p=\operatorname{char}k, k$ is the residue class field of $K$. Then $\alpha^m=c\varepsilon, c \in K^*, \varepsilon \in \mathcal O_L^*$. Since the residue class field extension is trivial, we may write $\varepsilon=uk, u \equiv 1 \bmod \mathfrak P, k\in \mathcal O_K^*$. By Hensel's lemma $x^m-u$ has a solution $\beta\in \mathcal O_L^*$, and $\alpha\beta^{-1}$ is a radical of $K$.
In the proof of (7.7) it use the property that the residue class field extension of $L/K$ is separable, but I could not see it is true for prop 7.11.
Is the proof make sense? If not, how to prove the statement? Thanks for any help.
Best Answer
The idea is to consider large powers of $p$ (the residue characteristic) to “kill” the “inseparable part” so that we can apply Neukirch’s argument again (I agree that the proof from the book seems incomplete).
Formally, consider $\alpha_t=\alpha^{mp^{t\varphi(m)}}$ for $t \geq 0$: then $\alpha_t=c^{p^{t\varphi(m)}}\varepsilon^{p^{t\varphi(m)}}$, $c \in K^{\times},\varepsilon \in O_L^{\times}$. Write $\alpha_t=c_tu_t$, $c_t \in K^{\times}$, $u_t \in O_L^{\times}$. If $t$ is large enough, then $u_t=r_tI_t$, with $I_t-1 \in \mathfrak{P}$, and $r_t \in O_K^{\times}$.
By Hensel, let $\beta_t \in O_L^{\times}$ be such that $\beta_t^m=I_t$. Then $R_t=\alpha^{p^{t\varphi(m)}}/\beta_t$ is a radical of $K^{\times}$, and $m|p^{t\varphi(m)}-1$ so that $R_t$ and $\alpha$ have the same image in $w(L^{\times})/v(K^{\times})$.