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

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Class field theory of imaginary quadratic fields

Can someone give me a good source that deals in detail with the Class Field Theory of imaginary quadratic fields? The sources on CFT that I have at hand only deal with CFT in general and then proceed ...
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$L$-function, easiest way to see the following sum?

What is the easiest way to see that$$\sum_{(m, n) \in \mathbb{Z}^2 \setminus \{0, 0\}} (m^2 + n^2)^{-s} = 4\zeta(s)L(s, \chi)?$$Here $\chi$ is the homomorphism $(\mathbb{Z}/4\mathbb{Z})^\times \to ...
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Quadratic field, $O_K/\mathfrak{p} = \mathbb{F}_p$, $O_K/pO_K$ is a finite field of order $p^2$.

Let $K$ be a quadratic field $\mathbb{Q}(\sqrt{m})$ where $m$ is a square free integer, and let $p$ be a prime number which does not divide $2m$. Where can I find a reference to a proof of the ...
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What does Lang mean here by “the usual criterion”?

Let $K$ be a number field containing the $n$th roots of unity, $S$ a finite set of places containing all the archimedean ones and all those which divide $n$, $K_S$ the group of $S$-units of $K$, and ...
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$K_S$ modulo $K_S^n$, where $K_S$ is the group of $S$-units

Let $K$ be a field containing the $n$th roots of unity, $S$ a finite set of places containing all the archimedean ones, and $K_S$ the group of $S$-units, i.e. those $x \in K^{\ast}$ which are units at ...
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On the solvability of the negative Pell equation $x^2-2py^2 = -1$

Given prime $p=8n+1$. Then $$x^2-2py^2 = -1\tag1$$ is not solvable for, $$p_1= 17, 73, 89, 97, 193, 233, 241, 257, 281, 337, 353, 401, 433, 449, 577, 593,601, 617, 641,\dots$$ but is solvable ...
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The class number and the inverse Galois problem

Let $G$ be finite group and $k$ a field. Inverse Galois theory asks if there is a galois extension $L/k$ such that $Gal(L/k) \simeq G$. Lets assume $k=\mathbb{Q}$ and let $\mathcal{h}_L$ denote the ...
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Blichfeldt-Minkowski Lemma

I'm trying to understand a proof of the following result Theorem: Let $K$ be a number field, and $|| \cdot ||$ the idelic norm (product of the normalized absolute values at each place). There ...
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Show an intersection of Galois groups is trivial

Let $L/K$ be a finite abelian extension of number fields, and for an extension of places $w/v$ consider the local Artin map $\Phi: K_v^{\ast} \rightarrow Gal(L_w/K_v)$, defined via the global Artin ...
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Relatives of Heegner numbers?

It is well known that Euler's lucky numbers are related to the Heegner numbers, where \begin{align} &n^2+n+p\\ \end{align} gives primes for $n=0,\dots,p-2$ if and only if its discriminant ...
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Difference between the Artin symbol and the Frobenius element?

The title says it all really! I have been reading 'Primes of the form $x^2+ny^2$', by Cox, and in chapter 5 he introduces the Artin Symbol, which for a field extension, $L/K$ is the unique element, ...
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Hasse invariant of quaternions over $\mathbb{Q}_p$

I am trying to compute the Hasse invariant of the quaternion algebra over $\mathbb{Q}_p$. I denote this algebra by $H$, and I'm assuming $p\equiv 3\pmod{4}$. So, $\mathbb{Q}_p(i)$ is an unramified ...
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Maximal p-subgroup of inertia group.

We know from the theory that if $\mathbb{L}$ is a finite Galois extension of the local field $\mathbb{K}$ then the ramification group $G_1$ is a $p$-group where $p$ is the characteristic of the ...
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Can we find the 18 imaginary quadratic ffields with class number 2 algorithmically?

I am reading about the class number problem. There is a well known complete list of imaginary quadratic fields $\mathbb{Q}(\sqrt{-d})$ with class number $1$. I found a paper by Stark that says ...
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Which properties determine the uniqueness of the local Artin map?

Any abelian extension of local fields can be realized as the completion of a global abelian extension. So let $L/K$ be abelian, $w/v$ an extension of places. From the global Artin map on ideles we ...
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Why is $m \infty$ the conductor of $K = \mathbb{Q}(\zeta_m)/\mathbb{Q}$?

Wouldn't this be saying that for all $p$ dividing $m$, $1 + p^{\operatorname{ord}_p(m)} \mathbb{Z}_p$ is contained in the group of local norms $N_{\mathfrak p/p}(K_{\mathfrak p})$, where $\mathfrak p$ ...
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Serge Lang ANT Theorem 7 p. 148

Let $\mathfrak c, \mathfrak f$ be admissible cycles for an extension of number fields $K/k$ with $\mathfrak f$ dividing $\mathfrak c$. If you're not familiar with the terminology, a cycle $\mathfrak ...
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Reference request: Cohomology of Elliptic Curves

Is it true that the group $$H^1(Gal(K^{ab}/K)/\mu_{\nu}(Gal(K_{\nu}^{ab}/K_{\nu})),E_{p^n})$$ is always p-divisible? Or are there any conditions which, when satisfied, guarantee its p-divisibility? ...
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$CL(O_S) \cong \mathbb{Z}/3\mathbb{Z}$.

Let $F = \mathbb{Q}(T)$ and let $X$ be the set of all places of $F$, and let $S = \{w\} \subset X$ where $w$ is the place of $F$ corresponding to the maximal ideal $(T^3 - 2)$ of $\mathbb{Q}[T]$. Let ...
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Artin reciprocity theorem for Hilbert class field

In Cox's book "Primes of the form $x^2 + ny^2 $..." gives the following statement of Artin reciprocity theorem, for the Hilbert class field (i.e. maximal unramified Abelian extension) Artin's ...
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Why is $Gal(\Omega/k)$ a topological group under the Krull topology?

For an infinite Galois extension $\Omega/k$ the Krull topology on $G:=Gal(\Omega/k)$ is defined by taking as a basis for the neighbourhood of an element $\sigma \in G$ all cosets of the form $\sigma ...
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How does $ \text{Gal}(K / k) $ act on ideles?

Let $K/k$ be cyclic of degree $N$, Galois group $G$. I want to define some action of $G$ on the group of ideles $J_K$ which commutes with multiplication. A natural way to do this is to take each ...
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How does class field theory help us deduce the splitting of nonprincipal prime ideals?

I had a general question about the significance of global class field theory. One of the goals, as I understand, is to answer the following question: Given $L/K$ abelian, $g$ a divisor of $[L : ...
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norm map and local class field theory

Let $K$ be a local field, say a finite extension of $\mathbb{Q}_p$ (which is the purpose of my interest). Let $L$ be an unramified extension of $K$. Local class field theory asserts that there ...
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prime ideal of $\mathcal O_K$ splits completely in tower of Galois extensions

The following exercise is taken from D.A.Cox's book "Primes of the form $x^2 +ny^2...$" He uses this in order to prove some intermediate steps before the proof of the main theorem in chapter 5, which ...
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definition of Artin map

Let $L/K$ be an unramified Abelian extension. Then the Artin symbol $ \left ( (L/K) / \mathfrak p \right) $ is defined for all prime ideals $\mathfrak p$ of $\mathcal O_K$. (because in an Abelian ...
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What does it mean for a prime ideal to divide a natural number m?

In Cassels and Frohlich (Algebraic Number Theory) Exercise 1, one is asked to derive some properties of the power residue symbol. It begins by stating the following: Let $m$ be a fixed natural ...
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Finite dimensional central division algebras over a finite extension of $\mathbb{F}_q(T)$

Over number fields, finite dimensional central division algebras are always cyclic algebras. So the construction of cyclic algebras is a nice recipe to create algebras, which exhausts all finite ...
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Existence of Conductor for Cyclotomic Extension (pg 200, Serge Lang A.N.T.)

Let $\zeta$ be a primitive $m$th root of unity, $m \not\equiv 2 \pmod 4$, and $K = \mathbb{Q}(\zeta)$. For $p$ prime and unramified i.e. $(m,p) = 1$, I know that the Artin symbol $(p, K/\mathbb{Q})$ ...
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The maximal unramified extension of a local field may not be complete

While reading my notes of a course in local class field theory, I arrived to a remark where it is said that given a complete discrete valuation field $K$, its maximal unramified extension $$K^{ur}= ...
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Unramified cubic extension of imaginary quadratic fields

Let $ K=\mathbb Q(\sqrt{-m})$ be an imaginary quadratic field with class number $ 6.$ Then by Hilbert class field theory and Galois correspondence it is known that $ K$ has a unramified cubic ...
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definition of the L-function $L(f, \chi, s): \mathbb{A}_K \rightarrow \mathbb{C}$, what is smoothness and what is $f$?

To summarize the question I'm going to ask: for those who have studied L-functions and class field theory, I am confused about the definitions of some things and haven't found a good reference for ...
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Discriminant of a Polynomial over a Local Field

I am trying to prove the local Kronecker-Weber theorem for tamely ramified abelian extensions $L|\mathbb{Q}_p$. At some point in the proof I need to show that $\mathbb{Q}_p(u^{1/e})$ is unramified ...
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Defining the Artin Map on the Ideles

Let $L/K$ be abelian. There is a natural way to define the Artin reciprocity map on the ideles using the notion of an admissible cycle. I don't want to go into the details of what that is right now, ...
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Algebraic proof of 2nd inequality of Global class field.

s there any 'explicit' proof, preferably without the use of cohomology, of second inequality of global field? By second inequality I mean $[C_K:N_{L/K}C_L]\leq [L:K]$ where $C_K$ is idele class group ...
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Where have I gone wrong in calculating norm residue symbol.

It is well known that the reciprocity map in local class field theory gives reciprocity map in Global class field theory. Namely, if $(\cdot,L_\beta/K_P):K_P^\times \rightarrow G(L_\beta/K_P)$ is the ...
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Are the discriminant of abelian cubic extensions of $\Bbb Q$ equal to the square of their conductor?

Here the conductor $N$ of an abelian extension $\Bbb Q \subset K$ is the smallest positive integer $N$ such that $K \subset \Bbb Q(\zeta_N)$. Thanks to class field theory there is an equivalence ...
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Does Idele group of norm 1 preserved by the norm?

I should explain my question in detail as of now I'm sure it makes no sense. Let $K$ be a global field (in particular I care about the characteristic $p$ case.) Then its Idele group $I_K$ has a ...
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Proof for a theorem on Cohomology by Tate

I am searching for a reference for the proof of the following theorem. Let $G$ be a finite group, let $C$ be a $G$-module, and let $u$ be an element of $\hat{H}^2(G,C)$. Assume that $\hat{H}^1(H,C) = ...
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Definition of module

In the book's, The theory of numbers, S. Iyanaga. Chater I, Cohomology of groups. What is the meaning of "module A"? Thank you all.
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Why do we have to work to prove the surjectivity of the local Artin map (Serge Lang A.N.T., Chapter XI Theorem 3)

I must be misunderstanding something about Artin reciprocity. Let $K/k$ be an abelian extension of number fields with Galois group $G$, $I_k$ the ideles of $k$, and $P$ a prime of $k$ (with $v$ a ...
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An interesting table of Prime Generating polynomials similar to $n^2+n+41$?

Here is some data on quadratic prime generating polynomials of a particular form. Kindly look at the questions given below it. $$\begin{array}{cccc} \text{#} & P(n)=an^2+bn+c\,; & d = ...
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Exercise 2.8 Cassels and Frohlich

I don't understand the discussion in exercise 2.8 of Cassels and Frohlich (page 352) beginning with "more generally". Why should it matter whether the formula for $c$ has a power of $-1$ in it if this ...
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Class number 1 for negative integers

If n is a positive integer then, $h(-4n) = 1 \iff n = 1,2,3,4,7$. The only proof I have seen of this is long and case wise. Is there any conceptual a priori reason to believe that these numbers are ...
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Is the group $I_K/K^{\ast}$ compact?

I have two question on adeles and ideles: $1)$Let $K$ be a number field. Is the group $I_K/K^{\ast}$ compact? Here $I_K$ is the idele group of $K$. $2)$ Also it will be helpful if someone explains ...
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Connected component of the Idele group

Let $K$ ba a number field with $r_1$ real embeddings and $r_2$ pairs of complex embeddings. Let $I_K$ be the group of ideles of $K$ and let $H$ be the connected component of identity. How to show that ...
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Reference request for Fourier analysis on local fields

I am studing Class field theory. I need a good reference books, notes e.t.c which explains the following topics : Ideles and ideals, haar volume measure and integration on local fields, Fourier ...
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Pure Cubic Fields

Let $K=\mathbb Q(\sqrt[3]m) $, $m $ cube free is a pure cubic field. Is there any relation between number of unramified quadratic extensions of $K$ and $2$ rank of the this field? Any help in this ...
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Generalized class group of $\mathbb Q(\sqrt{-5})$

I follow the notation of Georges Gras: Class Field Theory, some of which I recall for convenience; feel free to skip the following lines if you are familiar with the notation. Let $K$ be a number ...
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Cube roots of five

This is not really homework. I might be able to do this myself in time, from the methods in Ireland and Rosen. Note that every number has exactly one cube root $\pmod q$ for any prime $q \equiv 2 ...