Questions tagged [class-field-theory]

Class field theory is a major branch of algebraic number theory that studies abelian extensions of global and local fields.

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what was/is motivation and short history/story behind “class number”?

We know that there are some formula to calculate class number of a quadratic field $Q(\sqrt{d})$, where $d\in Z-\{0,1\}$, in terms of Dirichlet L-functions. Please correct me if I am wrong: for ...
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1answer
139 views

Discriminant of the ring class field of the order $\mathcal{O}=\mathbb{Z} \left[\frac{D+\sqrt{D}}{2}\right]$

Let's look at the procedure in the main theorem of this MO post: $\color{Red}{\text{Starting}}$ from a discriminant $D$, and at the $\color{Green}{\text{end}}$, we $\color{Green}{\text{find}}$ a ...
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$K^{*n}$ has finite index in $K^*$ for an infinite extension field $K$ of $\mathbb{Q}_p$?

Let $K\subset\mathbb{C}_p$ be an extension field of $\mathbb{Q}_p$, then when $K/\mathbb{Q}_p$ is a finite extension, then $K^{*n}$ has finite index in $K^*$(see this question, we can decompose $K^*$ ...
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Multiplicativity of Artin reciprocity in Neukirch's ANT

My question is about Lemma 5.4 in chapter 5 of Neukirch's ANT. The statement and proof are below. I think someone asked the exact same question some years ago but didn't get an answer, so I will try ...
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Lift Frobenius generator of Galois group $Gal(l/k)$ of an unramified extension $L/K$

We consider a finite unramified Galois field extension $L/K$ of non-Archimedean local fields with finite residue fields $l / k$. firstly, some notations: denote by $q \in O_K$ a uniformizer of $O_K$ ...
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1answer
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Class Field Towers

Let k a number field whose $p$-class group $C_k$ of type $(p,p)$ i.e $C_k \cong \mathbb{Z}/p\mathbb{Z}\times \mathbb{Z}/p\mathbb{Z}$. Let $k^{(1)}$ be the Hilbert class field of $k$ and $k^{(2)}$ the ...
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Algebraic closure of Puiseaux field $K((T))$ equals $\bigcup_{n \ge 1} K((T^{1/n}))$

I want to show that the algebraic closure $L:= \overline{K((T))}$ of Puiseaux field $K((T))$ for $K$ alg. closed of char $K=0$ equals the union $\bigcup_{n \ge 1} K((T^{1/n}))$. The closure clearly ...
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Group Isomorphism Question

In this pdf (https://kconrad.math.uconn.edu/blurbs/gradnumthy/classgroupKronecker.pdf) the author claims that the ideal class group of the ring of integers of $\mathbb{Q[\sqrt{-199}]}$ is the cyclic ...
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ord map from a local field to $\mathbb{Z}$

I'm reading J. Milne's notes on class field theory and near in the section of the chapter on local CFT he speaks of an exact sequence $$1 \to U_L \to L^{\times} \xrightarrow{\textrm{ord}_L} \mathbb{Z} ...
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Commutativity in local class field theory

Let $K$ be a finite extension of $\mathbb{Q}_p$, and $L,M$ be finite Galois extensions of $K$ such that $L\subset M$. Let $\phi_L\in Z^2(\text{Gal}(L/K),L^\times)$ such that $[\phi_L]$ is a generator ...
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Logarithm and Lubin-Tate formal group

Let $K$ be a finite extension, by Milne's online note "class field theory", $m_{\mathbb{C}_p}$ has a natural $O_K$ module structure where the action is given by $[a]_f$. For such a $f$, there exists a ...
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38 views

$x^2+14y^2$ mod $p$

Playing around with python, I noticed that for all prime $7\ne p<1000$, $x^2+14y^2$ covers all the residue classes in $(\mathbb{Z}/p\mathbb{Z})^{\times}$. I wondered if it is true for all $p\ne 7$, ...
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Dirichlet Class Number Formula Proof

I'm studying Davenport's proof of Dirichlet Class Number Theorem and I'm having troubles with this statement: Consider the sum $$\sum_{m_1 m_2<N; (m_1,m_2,d)=1} \left(\frac{d}{m_1}\right)= \...
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Proof of Shapiro's lemma with injective resolutions

I am reading Milne's CFT notes and am confused by his proof of Shapiro's lemma. Shapiro's lemma states that $H^r(G, Ind_H^G(N))\cong H^r(H, N)$. Milne's proof first shows that $N^H\cong Ind_H^G(N)^G$...
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Prime splitting in imaginary squarefree quadratic field

This appeared on my number theory final. I couldn't figure it out. The setup is: Let $K = \mathbb{Q}[\sqrt{-d}]$ be a quadratic number field, with $d = p_1...p_r$ a product of $r$ distinct odd ...
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The relation between the Chebotarev density theorem and Artin $L$-series

In ch. VII, §13, pp. 545-546 of the English translation of Algebraic Number Theory by J. Neukirch, one finds a 1-page proof of the Chebotarev density theorem. This proof seems fairly harmless, and ...
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Books for Tate's Thesis

I'm recently reading 'Fourier Analysis on Number Fields ',by Dinakar Ramakrishnan, Robert J. Valenza.The book is aiming to explain Tate's thesis,which is of my current interests.But I've found numbers ...
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Discriminant ideal of field extension $n(1+n)$ exponent

If $L/K$ is a degree $n$ number field extension, then a statement in a book says that if $L/K$ is unramified outside a finite set $S$ of prime ideals in $B=\mathcal{O}_K$, then the discriminant, $D(L/...
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Class-Field-theory

I am currently on Jürgen Neukirchs Algebraic Numbers theory and have some problems understanding his non-cohomological approach on class field theory. To introduce the notation would cost tremendous ...
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Why should we not consider $\mathbb{Q}(t)$ to be a global field?

I'm trying to properly understand the definition of the global field. I understand that there are two typical definitions of the global field: 1) $K$ is a global field if it is either a finite ...
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Number of Abelian Cubic Number Fields of given Discriminant

According to Cohn's paper on "the density of Abelian cubic fields", there are exactly $2^v$ Abelian cubic extensions of $\mathbb{Q}$ with discriminant $f^2$, where $v\ge0$ and $f$ is of the form $f=...
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Global Tate Duality Exercise in Neukirch

For $K$ a $\mathfrak p$-adic number field, local Tate duality yields a non-degenerate pairing $$H^1(K, \Bbb Z/n\Bbb Z) \times H^1(K, \mu_n) \longrightarrow \Bbb Z/n\Bbb Z,$$ where $\mu_n$ is the ...
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63 views

Ideal Class Group of $\mathbb{Q}(\sqrt{m})$ [closed]

I am trying to understand the ideal class group. I have seen a few different derivation of this. Sometimes i read about the ideal class group of a ring of algebraic integers ($\mathcal{O}_K$) and ...
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113 views

Algebraic closure of $k((t))$

Let $k$ be an algebraic closed field of characteristic $0$. I want to understand why the algebraic closure of the field $k((t))$ is $\underset{n\geq 1}{\bigcup k((t^{1/n}))}$. Obviously, $\underset{...
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1answer
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On the ambigous classes of quintic Kummer extension

Let $\Gamma = \mathbb{Q}(\sqrt[5]n)$ a pur quintic field and $k= \mathbb{Q}(\sqrt[5]n,\zeta_5)$ its normal closure. Let $\operatorname{Gal}(k/\Gamma)= \langle\tau\rangle$ with $\tau^4 = 1$ and $\...
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Places of an algebraic number field

I have been doing my best to learn the rudiments of class field theory via the formulation in terms of ideals. The $\textbf{ray class group}$ of an algebraic number field $K$ with respect to $\...
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Norm Group of unramified extension of certain degree

I am going through Neukirch's algebraic number theory, when I stumbled upon this result, which is not too clear to me. Here take $L| K$ to be a finite abelian extension of local fields, $\pi$ to be ...
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proof of Gal($\mathbb{Q}^{ab}/\mathbb{Q}$) $\rightarrow$ $\hat{\mathbb{Z}}^×$ [duplicate]

I want to prove that Gal($\mathbb{Q}^{ab}/\mathbb{Q}$) $\rightarrow$ $\hat{\mathbb{Z}}^×$ is an isomorphism by using the statement of class field theory Gal($L/K$)$^{ab}$ $\rightarrow$ $I_K/N_{L/K}...
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Artin map and ideles

I was going through Silverman's Advanced topics in the arithmetic of elliptic curves, and this statement was given without proof. Any hints/clarification would be appreciated. He first introduces the $...
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Finiteness of class number, Dirichlet's unit theorem and CFT

I want to study CFT and Milne's notes and Cassels are the two main sources. Milne seems to have all the ANT in Cassels plus two main theorems: finiteness of the class number and Dirichlet's unit ...
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115 views

class group size of cyclotomic field subextension

In the following, let $\mathbb{Q_1}$ denote the subfield of degree $p$ over $\mathbb{Q}$ in the $p^2$- cyclotomic extension. What is the best known upper bound for the size of its class group, $\text{...
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Inertia group of an extension of Local Fields

I am going through a proof in Chapter V of Neukirch's Algebraic Number theory, and would require some clarifications. The theorem we are trying to prove is that the local version of the Kronecker-...
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Index of norm group is degree of field extension

In Milne's class field theory, in the course of proving a corollary of local cft he states: "Since the index of a norm group is the degree of the abelian extension defining it..." Why is this ...
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Prerequisites for Serre's Algebraic Groups and Class Fields

What are the prerequisites for reading and understanding the book Algebraic Groups and Class Fields by Serre. Could you suggest some books to learn the prerequisites? Thanks
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Herbrand's unit theorem

Let $L / K$ be Galois extension of number fields with degree $n$, $U_{K},U_{L}$ be an unit group of an integer ring $\mathcal{O}_K,\mathcal{O}_L$ of $K,L$. $\sigma_1\dots \sigma_{r_K}$ are real ...
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102 views

a certain finite index subgroup of an unit group of a number field

I don't understand the proposition 2.3 of Gerald J.Janusz "Algebraic Number Fields" in Chapter V. It states as follow (changed some expressions based on Daileda's note p.39); Proposition. Let $L / ...
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Why a particular choice of power series $f$ working for Lubin-Tate theory?

This is related to Iwasawa's Local Class Field Theory Chpt 3 and 4. Let $k$ be a local field and $K$ be maximal unramified algebraic extension of $k$. Set $\Omega$ the algebraic closure of $k$. Take $...
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Totally ramified extensions of local fields

I'm trying to solve two exercises from a former course on Local Fields & Class Field Theory. Here is the original exercise sheet. Definition a field extension $L/K$ of non-archimedian local ...
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Number of ideals of norm $100$ of the Kleinian integers

The question to begin with is: how many ideals of norm $100$ does the ring of integers of $K=\mathbb{Q}(\sqrt{-7})$ have? I know the following stuff for certain: $\Delta_K=-7$ and $\mathscr{O}_K=\...
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Clarification regarding Silverman's proof of the description of Hilbert class field of a quadratic imaginary field

I was reading the proof of the following fact from Silverman's Advanced Topics in the Arithmetic of Elliptic Curves (p. 122). The Hilbert class field of a quadratic imaginary field $K$ with ring of ...
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How to prove the existence and properties of the nondegenerate pairing?

Let $K$ be a finite extension of $\mathbb{Q}_p$. Let $K_{nr}$ be the maximal unramified extension of $K$ and $\Gamma=\mathrm{Gal}(K_{nr}/K)$. Let $\mu_n$ be the set of $n$-th roots of unity. Write $H^...
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Extensions of Local Fields and the Trace Map

I am now having some trouble dealing with the trace map in the extension $K/F$ of local fields. Here are some questions. To introduce the discriminant, we may have two ways: Consider the valuation ...
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1answer
63 views

Field theoretic problem in a proof of Artin reciprocity law

Let $L/K$ be a cyclic extension of number fields of degree $n$, $\zeta_m$ a primitive $m$ th root of unity. Artin's lemma Let $S$ a finite set of primes of $\mathbb{Z}$, $\mathfrak{p}$ a prime of $\...
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On the determination of ambiguous ideal class of the extension $\mathbb{Q}(\zeta_5,\sqrt[5]{m})/\mathbb{Q}(\zeta_5))$

let $L=\mathbb{Q}(\sqrt[5]{n},\zeta_5)$ and $K=\mathbb{Q}(\zeta_5)$ the $5^{th}$ cyclotomic fields, we now that $[L:K] = 5$ and $\textrm{Gal}(L/K) = \langle \sigma \rangle$ so we call $\mathcal{A}$ ...
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Why Can we Write the Ray Class Group as a Quotient of Monoids

In the book Elliptic Functions, Lang uses a definition for the ray class group of a number field that I'm not completely comfortable with. I'm trying to understand why his definition is justified. ...
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1answer
108 views

Class groups and Iwasawa Theory

In a paper by Iwasawa (On Cohomology Groups of Units in $\mathbb{Z}_p$-Extensions, 1983) the following two facts have been mentioned without any explanation (or examples). I do not see how to justify ...
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Books which contain a proof of Kronecker's Jugendtraum

I am looking for books which contain a proof of Kronecker's Jugendtraum i.e. the imaginary quadratic field version of Hilbert's twelfth problem. Though it seems to be solved even the case for CM ...
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A certain decomposition of the idèles group of a global field

I am currently reading the book "Algebraic number theory" published by Cassels and Fröhlich. Let $K$ be a global field (which we may think as a number field as far as I am concerned) and $L/K$ a ...
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elliptic curve analogue of ambiguous class number formula

The ambiguous class number formula (first proven by Chevalley) gives the number of (strongly) ambiguous ideal classes in terms of the class number $h(K)$ of the base field $K$, the number $t$ of ...
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161 views

discriminants in ring of integers

I've been trying to solve the following problem: Find the ring of integer of $\mathbb{Q}(\theta)$ when $\theta^3 + \theta + 1 = 0$. I started by computing the discriminant, which is $-31$. Then I ...

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