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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 $t \geq 0$:

$$ \mathfrak{R}_t = \lbrace x \in K^{\times} \mid x^{1-\sigma} \text{ is in } K^{\times p^t} \mu_p \rbrace $$

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?

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?

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 $t \geq 0$:

$$ \mathfrak{R}_t = \lbrace x \in K^{\times} \mid x^{1-\sigma} \text{ is in } K^{\times p^t} \mu_p \rbrace $$

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 have a general description in some cohomological and/or algebraic number-theoretic framework?

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 $t \geq 0$:

$$ \mathfrak{R}_t = \lbrace x \in K^{\times} \mid x^{1-\sigma} \text{ is in } K^{\times p^t} \mu_p \rbrace $$

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?

If $K/k$ is a degree $p$ Kummer extension of number fields (so $k$ contains the $p^r$th roots of unity for some $r \geq 1$), 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 $t \geq 0$:

$$ \mathfrak{R}_t = \lbrace x \in K^{\times} \mid x^{1-\sigma} \text{ is in } K^{\times p^t} \mu_p \rbrace $$

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 have a general description in some cohomological and/or algebraic number-theoretic framework?

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 $t \geq 0$:

$$ \mathfrak{R}_t = \lbrace x \in K^{\times} \mid x^{1-\sigma} \text{ is in } K^{\times p^t} \mu_p \rbrace $$

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 have a general description in some cohomological and/or algebraic number-theoretic framework?