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I am trying to work out why natural density implies Dirichlet density. So let $T$ be a subset of rational primes with natural density $\rho$, i.e. $\lim_{x\to +\infty}\frac{\pi_T(x)}{\pi(x)}=\rho$, wh …

I want to compute the discriminant of the number field $K:=\mathbb Q(\alpha)$, where $\alpha^5=2.$ I guess that $\{1,\alpha,\alpha^2,\alpha^3,\alpha^4\}$ is an integral basis of $\mathcal O_K,$ howeve …

$\newcommand{\O}{\mathcal O}$ $\newcommand{\p}{\mathfrak p}$ $\newcommand{\a}{\alpha}$ $\newcommand{\N}{\operatorname{N}_{L/K}}$ $\newcommand{\ol}{\overline}$ $L/K$ is an extension of number fields an …

Let $K$ be a number field, and let $k_n\in K^*$ such that $\{k_n\}_n$ converge to $a_w\in K_w^*$ for all $w\in V_K-\{v_0\}$, where $v_0$ is a given place of $K$, can I say that $\forall w\in V_K-\{v_0 …

I need to prove that $N_{L/K}(U_L^i)\supseteq U_K^i$, where $K$ is a $p$-adic number field, $L/K$ is a finite unramified field extension, and $$U_L^i=\begin{cases}\mathcal O_L^\times, &i=0,\\ 1+\mathf …

After discussing with others, I find out where I am wrong. The point is that the map $$N_i:U_L^i/U_L^{i+1}\to \mathcal O_L/\mathfrak m_L\to\mathcal O_K/\mathfrak m_K\to U_K^i/U_K^{i+1}, \overline{1+\p …

I need to compute $Cl(K)$ and $Cl^+(K)$ for the fields $K = \mathbb Q(\sqrt2)$ and $K = \mathbb Q(\sqrt3)$. Here $Cl(K)$ is the ray class group of trivial modulus, and $Cl^+(K)$ is the narrow class gr …

What's the meaning of $\mod (1+p^n\mathbb Z_p)$? I am learning $p$-adic numbers, and in one exercise of proving that $x\mapsto (1+p)^x$ defines an isomorphism of abelian groups $(\mathbb Z_p,+)\to (1+ …

Let $K$ be a complete discrete valuation field with an algebraically closed residue field $k.$ Let $m \ge 1$ be an integer prime to the characteristic of $k$. Let $\pi_K$ be an arbitrary uniformizer …