I am trying to understand Remark 7.2.22 (Page 256) of Algebraic Function Fields and Codes (Second Edition) by Henning Stichtenoth.
In that remark he considers a tower $\mathcal{F} = (F_0,F_1,F_2,\dots)$ over a finite constant field $\mathbb{F_q}$ and extends it constantly with $L$ to get $\mathcal{F'}=(F_0',F_1',F_2',\dots)$ (i.e. $L/\mathbb{F_q}$ is an algebraic extension (may be finite or infinite) and $F_i' = F_iL$ $\forall i \geq 1$).
Next he takes
a place $P \in \mathbb{P}(F_i)$ for some $i >0$ and asserts that $P$ ramifies in the extension $F_{i+1}/F_i$ if and only if the places $P' \in \mathbb{P}(F_i')$ lying above $P$ are ramified in the extension $F_{i+1}'/F_i'$.
My doubt here is if the situation is as given below
Case 1: $$ \begin{array}{ccccccc} F_i' & \rightarrow & F_{i+1}' &&P'&\rightarrow & P_1'\\ \uparrow & & \uparrow && \uparrow && \uparrow\\ F_i& \rightarrow & F_{i+1} && P &\xrightarrow{e>1} & P_1 \end{array} $$ (where $e$ is the corresponding ramification index in the diagram) then as $e(P'/P)=e(P_1'/P_1)=1$ we can say that $e(P_1'/P')=e(P_1/P) >1$.
But what if the situation as depicted in the following diagram also happens
Case 2: $$ \begin{array}{ccc} Q & \rightarrow & P_2'\\ \uparrow && \uparrow \\ P & \xrightarrow{e=1} & P_2 \end{array} $$ i.e. $P_1, P_2$ both extend $P$ with one extension ramified and another unramified. In this case $e(P_2'|Q)$ has to be equal to $1$.
How can he conclude that every place $P' \in \mathbb{P}(F_i')$ gets ramified if $P$ is ramified? Doesn't Case 2 situation arise?
We have the following definition (item (b) of Definition 3.5.4 in Stichtenoth's book):
Definition: Given an algebraic field extension $F'/F$, a place $P\in\mathbb P(F)$ is said to be ramified if $e(P'|P)>1$ for some place $P\in\mathbb P(F')$ above $P$ .
Let $F_i, F_i'$ as in your question, and let $P\in\mathbb P(F_i)$. We want to show that $P$ is ramified in $F_{i+1}/F_i$ iff $P'\in\mathbb P(F_i')$ is ramified in $F_{i+1}'/F_i$ for all $\boldsymbol{P'}$ above $\boldsymbol P$.
Suppose then that $P$ is ramified in $F_{i+1}/F_i$. Then there is a place $P_1\in\mathbb P(F_i')$ above $P$ such that $e(P_1|P)>1$. Let $P'$ be any place in $\mathbb P(F_i')$ above $P$. We want to find a place $P_1'\in\mathbb P(F_{i+1}')$ above $P'$ such that $e(P_1'|P')>1$.
We want to use the existence of the place $P_1$; more concretely, if we are able to find $P_1'$ simultaneously above $P'$ and $P_1$, then we are done, because in this case
$$e(P_1'|P')e(P'|P)=e(P_1'|P_1'\cap F_{i+1})e(P_1'\cap F_{i+1}|P)\,,\tag{$\ast$}$$
and since $e(P'|P)=e(P_1'|P_1'\cap F_{i+1})=1$ (constant field extensions), it follows that $e(P_1'|P')=e(P_1'\cap F_{i+1}|P)=e(P_1|P)>1$.
If $F_i'$ and $F_{i+1}$ were linearly disjoint over $F_i'\cap F_{i+1}$, then we can use the following result:
Lemma: Let $F_1,F_2$ be function fields over a perfect field $k$. Suppose that $F_1$ and $F_2$ are linearly disjoint over $F=F_1\cap F_2$, and let $P\in\mathbb P(F)$ and $P_i\in\mathbb P(F_i)$ for $i=1,2$ be places such that $P_1|P$ and $P_2|P$. If the compositum $F_1F_2$ is defined, then exists a place $Q\in\mathbb P(F_1F_2)$ such that $Q|P_1$ and $Q|P_2$.
This result appears as Lemma 2.1.3 in Jörg Wulftange's PhD Thesis, which can be found here. I strongly believe that this hypothesis is satisfied in your case, but I am too lazy now to think about it.
For the converse, if $P$ is unramified in $F_{i+1}/F_i$, then $e(P_1|P)=1$ for all $P_1\in\mathbb P(F_{i+1})$ above $P$, so for any $P'\in\mathbb P(F_i')$ above $P$ and any $P_1'\in\mathbb P(F_{i+1}')$ above $P'$ we have $e(P_1'|P')=1$ (do the same reasoning as in $(\ast)$ above).