# 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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### 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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### 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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### 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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### 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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### 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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### 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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### 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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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 $\mathscr{O}_K=\... 1answer 44 views ### Clarification regarding Silverman's proof of the description of Hilbert class field of a quadratic imaginary field I was reading the proof of the following fact from Silverman's Advanced Topics in the Arithmetic of Elliptic Curves (p. 122). The Hilbert class field of a quadratic imaginary field$K$with ring of ... 0answers 18 views ### How to prove the existence and properties of the nondegenerate pairing? Let$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^...
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 ...