Let $F$ be an irreducible squarefree cubic polynomial over a number field $K$. Let $L:=K[x]/{(F(x))}$ be a cubic extension of $K$.
Suppose that $\alpha \in L^\times$ such that $N_{L/K}(\alpha)=\beta^2$ for some $\beta \in K^\times$, where $N_{L/K}$ is the norm map of the field extension.
Let $v$ be a finite place of $K$, and let $\{w_1, \cdots, w_n\}$ be the set of all places of $L$ dividing $v$ ($1\leq n \leq 3$).
Let $L_v:=K_v[x]/{(F(x))}$ be a $K_v$-algebra. Then we have $L_v \simeq L_{w_1} \times \cdots \times L_{w_n}$. Let $(\alpha_1, \dots, \alpha_n)$ be the image of $\alpha \in L^\times \hookrightarrow L_v \simeq L_{w_1} \times \cdots \times L_{w_n}$. Here is my question: Can we always have
$$ \text{val}_{w_i}(\alpha_i) \in 2\mathbb{Z} \text{ for all } 1\leq i \leq n? $$
If $F(x)$ is also irreducible over $K_v$, then $L_v$ is a field and $\text{val}_v(N(\alpha))=f \times \text{val}_{w_1}(\alpha_1)$, where $f$ is the inertial degree of $L_{w_1}/K_v$. Since $\text{val}_v(N(\alpha))=2\times \text{val}_v(\beta))$ is even and $f \in \{1, ~3\}$ we can obtain that $\text{val}_{w_1}(\alpha_1) \in 2\mathbb{Z}$. My question is that whether the above statement is true when $n=2$ or $3$.
