If $K/k$ is a degree $p$ Kummer extension of number fields (so $k$ contains the $p^r$th roots of unity for some $r \geq 1$ --- let's also assume $K/k$ is not generated by $p$-power roots of unity), I have seen the following group called the radical group of the extension:
$$ \mathfrak{R}_{\infty} = \lbrace x \in K^{\times} \mid x^p \text{ is in } k \rbrace $$
If we let $\sigma$ be a generator of $G$, the Galois group of $K/k$, and let $\mu_{p^r}$ be the group of $p^r$th roots of unity, we can give the alternative definition
$$ \mathfrak{R}_{\infty} = \lbrace x \in K^{\times} \mid x^{1-\sigma} \in \mu_{p^r} \rbrace $$
My first question: is there a standard name for the groups in the filtration, defined for $t \geq 0$:
$$ \mathfrak{R}_t = \lbrace x \in K^{\times} \mid x^{1-\sigma} \text{ is in } K^{\times p^t} \mu_{p^r} \rbrace $$
We have
$$\mathfrak{R}_0 = K^{\times}$$
and
$$\bigcap_{i=0}^{\infty} \mathfrak{R}_t = \mathfrak{R}_{\infty}$$
We can define two symbols on $\mathfrak{R}_t$. If $x$ is in $\mathfrak{R}_t$ and $x^{1-\sigma} = \delta^{p^t} \zeta$, then
$$ \left( \sigma, x \right)_{\mathrm{Kum}} = \zeta$$
and
$$ \left( \sigma, x \right)_{N} = N_{K/k} \delta $$
where $N_{K/k}$ denotes the field norm.
These symbols have the following properties:
1) Both $\left( \sigma, x \right)_{\mathrm{N}}^{p^{r-1}}$
and
$$\left( \sigma, x \right)_{\mathrm{Kum}} \left( \sigma, x \right)_{N}^{p^{t-1}}$$
are $p$th roots of unity and are independent of the decomposition $x^{1-\sigma} = \delta^{p^t} \zeta$.
2) If $t \geq r$, then $\left( \sigma, x \right)_{\mathrm{Kum}}$ is a $p$th root of unity and is independent of the decomposition $x^{1-\sigma} = \delta^{p^t} \zeta$.
3) The maps $\left( \sigma, x \right)_N^{p^{r-1}}$, $\left( \cdot, \cdot \right)_{\mathrm{Kum}} \left( \cdot, \cdot \right)_{N}^{p^{t-1}}$,
and, when $t \geq r$,
$$\left( \cdot, \cdot \right)_{\mathrm{Kum}}$$
are bihomomorphisms, so define elements of the cohomology group $H^1 (G, \mu_p)$.
4) The maps
$$\left( \sigma, \cdot \right)_{\mathrm{Kum}} \left( \sigma, \cdot \right)_{N}^{p^{t-1}}$$
and
$$\left( \sigma, \cdot \right)_{\mathrm{Kum}}$$
in 1) and 2) above respectively restrict to the usual Kummer pairing on $\mathfrak{R}_{\infty}$
My second question is, do the above filtration and symbols fit into some more general cohomological and/or algebraic number-theoretic framework?
