nt.number theory - Analogue of a ring extension splitting in the Kummer case

Background (the Kummer extension case)

Let $R$ be a complete regular local ring (it follws that it's a UFD) with a prime integer $p$ contained in the maximal ideal of $R$ (I'm mostly interested in $R=V[[x_1ldots x_n]]$ with V a DVR). Consider a finite extension of rings $Rhookrightarrow A$ with $A = R[u^{frac{1}{p}}]$.

Let $K$ and $F$ be the fraction fields of $R$ and $A$ respectively. Let $xi$ be a $p^{th}$ primitive root of 1 and consider the following commutative diagrams where horizontally displayed extensions are of order $p-1$ and vertical ones are of order $p$.

$begin{array}{ccc}
K & hookrightarrow & K[xi]\
downarrow & & downarrow \
L &hookrightarrow & L[xi]\
end{array}$
where $K[xi]subset L[xi]$ is a Kummer extension with $Gal((L[xi]/K[xi])=<sigma>simeq mathbf{Z}/pmathbf{Z}$

$begin{array}{ccc}
R & hookrightarrow & R' \
downarrow & & downarrow \
A &hookrightarrow & A' \
end{array}$
where $R'$ is the integral closure of $R$ in $R[xi]$ and $A'$ is the integral closure of $A$ in $L[xi]$.

Now $L[xi]$ has an eigenspace decomposition $L[xi]=L_1bigoplus ldots bigoplus L_{p} $, where $L_i= { x in L[xi]| sigma(x)=xi^i *x }$.

Here the contractions $S_i=A'cap L_i$ are rank one $R'$-modules (in fact $ S_i= {x in A' |sigma(x)=xi^i*x}$).

Question

Suppose that we DROP the hypothesis $A = R[u^{frac{1}{p}}]$ and only assume $[L:K]=p$. Let $L'$ be such that $L'/K$ is Galois and consider the analogous diagrams
$begin{array}{ccc}
K & hookrightarrow & K'\
downarrow & & downarrow \
L &hookrightarrow & L'\
end{array}$ and
$begin{array}{ccc}
R & hookrightarrow & R'\
downarrow & & downarrow \
A &hookrightarrow & A'\
end{array}$.

$L'$ still has an eigenspace decomposition of the type $L'=L_1bigoplus ldots bigoplus L_{p} $, where $L_isimeq L$.

What conditions would ensure that the contractions $S_i=L_icap A'$ are still rank one $R'$-modules?

I am guessing that we need $[L':L]$ not to be divisible by $p$, but would be happy to see a proof even under more restrictive conditions (such as $Gal(L'/L)$ commutes with $Gal(L'/R')$ perhaps).

Any help is much appreciated.