map for the extension of fields $L_{K,i}/K$.

In order to ease notation, we establish the following notational conventions: when $K={\mathbb Q}$ we will drop the field from the subscripts altogether, and if $K={\mathbb Q}_v$, we will just use the subscript $v$. This convention will apply to *anything* that has a field as a subscript throughout the rest of the paper. As an example, $L_v={\mathbb Q}_v[T]/(f(T))$ and $L={\mathbb Q}[T]/(f(T))$.

Following standard notational conventions, we let ${\mathcal{O}}_K,\ I(K),$ and ${{\rm Cl}}(K)$ denote the ring of integers of $K$, the group of fractional ideals in $K$, and the ideal class group of $K$, respectively. We define analogous objects for the algebra $L_K$ as products of each component, as follows: $$\begin{aligned}
{\mathcal{O}}_{L_K}&={\mathcal{O}}_{L_{K,1}}\times\cdots\times{\mathcal{O}}_{L_{K,m_K}},\\
I(L_K)&=I({L_{K,1}})\times\cdots\times I({L_{K,m_K}}),\\
{{\rm Cl}}(L_K)&={{\rm Cl}}({L_{K,1}})\times\cdots\times {{\rm Cl}}({L_{K,m_K}}).\end{aligned}$$

\[defn-res\] Let $K$ be a field extension of ${\mathbb Q}$, and let $L={\mathbb Q}[T]/(f(T))$ be as before.

1.  Let $I_v(L)$ denote the subgroup of $I(L)$ consisting of prime ideals in $L$ with support above a prime $v$ in ${\mathbb Z}$. For a finite set $S$ of places of ${\mathbb Q}$, let $$I_S(L)=\prod_{v\in S\setminus\{\infty\}} I_v(L).$$

2.  For any field extension $K$ of ${\mathbb Q}$, let $$H_K = \ker\left( N_K\colon L_K^\times/(L_K^\times)^2\to K^\times/(K^\times)^2 \right).$$ For any place $v$ 