The following is the statement from Algebraic Number Theory by Neukirch (Chapter 2 Proposition(7.7) p155)
Blockquote Suppose $K$ is Henselian field, $p=char(\kappa)$ , the character of the residue field of $K$ . A finite extension $L/K$ is tamely ramified if and only if the extension $L/T$ , ($T$ is the maximal unramified subextension of $L/K$ ) is generated by radicals $L=T(\sqrt[m_1]{a_1}\dots \sqrt[m_r]{a_r})$ , such that $(m_i,p)=1$ .
For "$\Rightarrow$" direction, the proof given in the book is correct, but it should be pointed out that "$a_i$"s come from $T$ .
The proof of "$\Leftarrow$ " direction is highly suspicious. First of all, what's the right statement? There are at least two ways:
(1) $K$ is a Heselian field, for $a_i \in K$ Let $L=K(\sqrt[m_1]{a_1}\dots \sqrt[m_r]{a_r}) \qquad (m_i,p)=1 \qquad p=char(\kappa)$ . Then $L/K$ is a tamely ramified extension.
(2) Same as (1) + $K$ is just the maximal unramified subextension of $L/K$ (i.e. $L/K$ is totally ramified ).
Does anyone know the proof of either statement? In addition, if $L/K$ happens to be a finite Galois extension (or maybe you only need simple extension), is it true $L=\sqrt[m]{a}$ form?