Let $L_0$ be a global field without real places, that is, a global function field or a totally imaginary number field, and let $V_f(L_0)$ denote the set of finite (that is, non-archimedean) places of $L_0$.
Let $n>1$ be a natural number. For a given finite set $S_0\subset V_f(L_0)$, we construct a cyclic extension $L_1/L_0$ of degree $n$ such that $$(L_0)_{v_0}\otimes_{L_0} L_1 \text{ is a $field$ for all places $v_0\in S_0\subset V_f(L_0)$}.$$ Here $(L_0)_{v_0}$ denotes the completion of the field $L_0$ at the place $v_0$. This construction is possible by Theorem X.6 (which I use as a black box) in the book "Class Field Theory" by Artin and Tate. Moreover, this construction is not unique: we make choices.
Let $r>1$ be a natural number. We choose a finite set $S_1\subset V_f(L_1)$ according to a certain rule, and construct a cyclic extension $L_r/L_1$ of degree $n^{r-1}$ such that $$(L_1)_{v_1}\otimes_{L_1} L_r\ \text{ is a $field$ for all places $v_1\in S_1\subset V_f(L_1)$}.$$ Again, we use Theorem X.6 of Artin-Tate, and again our construction is not unique: we make choices.
Then the extensions $L_r/L_1$ and $L_1/L_0$ are cyclic, hence normal, but the extension $L_r/L_0$ might not be normal.
Question 1. Can we make good choices when constructing the cyclic extensions $L_1/L_0$ and $L_r/L_1$ so that the extension $L_r/L_0$ will be normal?
Now let $\sigma$ be a generator of the Galois group ${\rm Gal}(L_r/L_1)$. Then the fixed subfield in $L_r$ of $\sigma^{n^0}$ is $L_1$, and the fixed subfield in $L_r$ of $\sigma^{n^{r-1}}$ is $L_r$. For $1\le s\le r$ we define $L_s$ to be the fixed subfield in $L_r$ of $\sigma^{n^{s-1}}$; then the new definition of $L_1$ and $L_r$ coincides with the old one.
The extension $L_s/L_1$ is a cyclic extension of degree $n^{s-1}$, hence normal, and the extension $L_1/L_0$ is normal too.
Question 2. Can we make good choices when constructing the cyclic extensions $L_1/L_0$ and $L_r/L_1$ so that the extensions $L_s/L_0$ will be normal for all $s$?
