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Ehsan Shahoseini
Ehsan Shahoseini
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Let $K$ be a number field, $\mathfrak{p}$ be a prime of it, and $L=K(\mathfrak{p}^n)$ be the ray class field of $K$ with finite conductor $\mathfrak{p}^n$ (we do not care about the infinite part of the conductor).

  1. Is it true that $L/K$ is totally ramified at $\mathfrak{p}$?

  2. (Here we may assume $L/K$ is cyclic.) Is it true that the order of the zeroth Tate cohomology group $\smash{\hat{H}}^0(\operatorname{Gal}(L/K), U_L)$ is equal to $2^t$, where $U_L$ is the group of units of the ring of integers of $L$ and $t$ is the number of infinite places of $K$ which ramifyramified in $L$?

Let $K$ be a number field, $\mathfrak{p}$ be a prime of it, and $L=K(\mathfrak{p}^n)$ be the ray class field of $K$ with finite conductor $\mathfrak{p}^n$ (we do not care about the infinite part of the conductor).

  1. Is it true that $L/K$ is totally ramified at $\mathfrak{p}$?

  2. (Here we may assume $L/K$ is cyclic.) Is it true that the order of the zeroth Tate cohomology group $\smash{\hat{H}}^0(\operatorname{Gal}(L/K), U_L)$ is equal to $2^t$, where $U_L$ is the group of units of the ring of integers of $L$ and $t$ is the number of infinite places of $K$ which ramify in $L$?

Let $K$ be a number field, $\mathfrak{p}$ be a prime of it, and $L=K(\mathfrak{p}^n)$ be the ray class field of $K$ with finite conductor $\mathfrak{p}^n$ (we do not care about the infinite part of the conductor).

  1. Is it true that $L/K$ is totally ramified at $\mathfrak{p}$?

  2. (Here we may assume $L/K$ is cyclic.) Is it true that the order of the zeroth Tate cohomology group $\smash{\hat{H}}^0(\operatorname{Gal}(L/K), U_L)$ is equal to $2^t$, where $U_L$ is the group of units of the ring of integers of $L$ and $t$ is the number of infinite places of $K$ which ramified in $L$?

`\smash`; `\operatorname`
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LSpice
LSpice
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Ramification of primes and order of $\hat$\smash{\hat{H}}^0$ in ray class fields with one finite prime divisor

Let $K$ be a number field, $\mathfrak{p}$ be a prime of it, and $L=K(\mathfrak{p}^n)$ be the ray class field of $K$ with finite conductor $\mathfrak{p}^n$ (we do not care about the infinite part of the conductor).

1-Is it true that $L/K$ is totally ramified at $\mathfrak{p}$?

2-(Here we may assume $L/K$ is cyclic.) Is it true that the order of the zeroth Tate cohomology group $\hat{H}^0(Gal(L/K), U_L)$ is equal to $2^t$, where $U_L$ is the group of units of the ring of integers of $L$ and $t$ is the number of infinite places of $K$ which ramify in $L$?

  1. Is it true that $L/K$ is totally ramified at $\mathfrak{p}$?

  2. (Here we may assume $L/K$ is cyclic.) Is it true that the order of the zeroth Tate cohomology group $\smash{\hat{H}}^0(\operatorname{Gal}(L/K), U_L)$ is equal to $2^t$, where $U_L$ is the group of units of the ring of integers of $L$ and $t$ is the number of infinite places of $K$ which ramify in $L$?

Ramification of primes and order of $\hat{H}^0$ in ray class fields with one finite prime divisor

Let $K$ be a number field, $\mathfrak{p}$ be a prime of it, and $L=K(\mathfrak{p}^n)$ be the ray class field of $K$ with finite conductor $\mathfrak{p}^n$ (we do not care about the infinite part of the conductor).

1-Is it true that $L/K$ is totally ramified at $\mathfrak{p}$?

2-(Here we may assume $L/K$ is cyclic.) Is it true that the order of the zeroth Tate cohomology group $\hat{H}^0(Gal(L/K), U_L)$ is equal to $2^t$, where $U_L$ is the group of units of the ring of integers of $L$ and $t$ is the number of infinite places of $K$ which ramify in $L$?

Ramification of primes and order of $\smash{\hat{H}}^0$ in ray class fields with one finite prime divisor

Let $K$ be a number field, $\mathfrak{p}$ be a prime of it, and $L=K(\mathfrak{p}^n)$ be the ray class field of $K$ with finite conductor $\mathfrak{p}^n$ (we do not care about the infinite part of the conductor).

  1. Is it true that $L/K$ is totally ramified at $\mathfrak{p}$?

  2. (Here we may assume $L/K$ is cyclic.) Is it true that the order of the zeroth Tate cohomology group $\smash{\hat{H}}^0(\operatorname{Gal}(L/K), U_L)$ is equal to $2^t$, where $U_L$ is the group of units of the ring of integers of $L$ and $t$ is the number of infinite places of $K$ which ramify in $L$?

Source Link
Ehsan Shahoseini
Ehsan Shahoseini
  • 333
  • 1
  • 8

Ramification of primes and order of $\hat{H}^0$ in ray class fields with one finite prime divisor

Let $K$ be a number field, $\mathfrak{p}$ be a prime of it, and $L=K(\mathfrak{p}^n)$ be the ray class field of $K$ with finite conductor $\mathfrak{p}^n$ (we do not care about the infinite part of the conductor).

1-Is it true that $L/K$ is totally ramified at $\mathfrak{p}$?

2-(Here we may assume $L/K$ is cyclic.) Is it true that the order of the zeroth Tate cohomology group $\hat{H}^0(Gal(L/K), U_L)$ is equal to $2^t$, where $U_L$ is the group of units of the ring of integers of $L$ and $t$ is the number of infinite places of $K$ which ramify in $L$?