Let $K$ be a local field over $\mathbb{Q}_2$ such that the extension $K(i)/K$ is ramified and let $U^1_{K(i)}$ and $U^1_K$ denote the groups of principal units in the fields $K(i)$ and $K$, respectively.
We write $(\cdot \phantom{.},\cdot)_{2, K} : K^\ast/(K^\ast)^2 \times K^\ast/(K^\ast)^2 \longrightarrow \mu_2$ for the $2$th power Hilbert symbol.
Now i want to show that for a unit $\eta \in U^1_K$ the following statement is hold:
$\eta \in N_{K(i)/K}U^1_{K(i)}$ if and only if $(\eta, \eta)_{2,K} = 1$
I´ve already shown the direction $"\Rightarrow"$ by direct calculation. For the direction $"\Leftarrow"$ i have the hint to show that the indices of the subgroups $N_{K(i)/K}U^1_{K(i)}$ and $\{\eta \in U^1_K | (\eta, \eta)_{2,K} = 1\}$ in $U^1_K$ are equal to $2$.
Unfortunately, i am not able to prove this. But i know that by local class field theory one has $K^\ast/N_{K(i)/K}K(i)^\ast \cong Gal(K(i)/K)^{ab} (\cong \mathbb{Z}/2\mathbb{Z})$ and i think that for the second subgroup it may be useful to know that $(\eta, \eta)_{2,K} = (\eta, -1)_{2,K}$.
