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

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Some property of norm residue symbol

On the last page of this article, Artin and Hasse used some property of the norm residue symbol $$\left(\frac{\lambda}{A}\right)_K = \left(\frac{\lambda}{n(A)}\right)_{k_\zeta}$$ (under Beweis von ...
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Relating the class number of a field, and of its normal closure

Suppose I take a number field $ K $, not necessarily Galois, with class number $ h_k $ (over $ \mathbb{Q} $). Write $ \overline{K} $ for the normal closure of $ K $. What, if anything, can be said ...
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Explicit correspondences in cohomology

$\newcommand{\Z}{\mathbb{Z}} \newcommand{\Q}{\mathbb{Q}}$ In class field theory, one typically identifies many groups such as $$\begin{align*} &H^{-2}(G, \Z) \cong H_1(G, \Z) \cong H_0(G,I_G) \...
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Image of the norm map in imaginary quadratic fields

Let $K=\mathbb{Q}(\sqrt{D})$ be an imaginary quadratic field of discriminant $D<0$. I want to know the image of the norm map $$ N^K_{\mathbb{Q}}:\mathcal{O}_K\to\mathbb{Z} $$ and the values of $N^...
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How to construct rings with a given class number?

Hi I was learning about class number and I was wondering if it is known how to construct rings for any specific class number.
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Explicit formula for Hilbert power residue symbol for extension of $\mathbb{Q}_p$

Artin and Tate's Class Field Theory provides an explicit computation of the local power residue symbol (Chapter XII, Theorem 10, page 119). Is there similar formulas for arbitrary extension of $\...
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Proof for Neukirch mysterious relationship between frobenius elements in abstract CFT

In Neukirch's ANT chapter (4) on Abstract Class Field Theory, there is a claim which I can prove, but I can't prove the "In particular" part that follows. I'm stuck in this for almost a week. I've ...
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How to find open subgroups of finite index in $\mathbb{Q}_{3}^{\times}$?

For purposes of illustrating Local Class Field Theory, let us play with the $3$-adic numbers. I'd like to find some open subgroups of finite index in $\mathbb{Q}_{3}^{\times}$. I know about the ...
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Stronger form of Hensel's lemma?

Let $f \in \mathbb{Z}_p[x]$ and suppose $|f(a)|_p < |f'(a)|_p^2$ for some $a \in \mathbb{Z}_p$. Let $a_1 = a$, and for $n \ge 1$ let$$a_{n+1} = a_n - f(a_n)/f'(a_n).$$How do I see that this defines ...
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Discriminant cusp form and generator of an ideal in the Hilbert class field (proof of prop. 14.1 Heegner points on $X_0(N)$, Gross)

Let $K$ be a quadratic imaginary field where the rational prime $N$ splits: $\mathfrak{n}\cdot\bar{\mathfrak{n}}=(N)$ and denote with $H_K$ the Hilbert class field of $K$. At the beginnig of the ...
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Complex multiplication and ray class fields

This question is mainly referring to the proof of Theorem 5.6, Chapter 2 of Silverman's "Advanced Topics in the AEC". Basically, let $K$ be an imaginary quadratic field, and $E$ be an elliptic curve ...
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Proof of Kummer's Lemma in S. Langs 'Cyclotomic fields'

I was going through the proof of Kummer's Lemma (stated below) as done in Serge Langs Cyclotomic fields on page 312. Now the author states that by class field theory it suffices to show that $\...
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On the integer solutions to $u^2+163v^2=w^3$ and others

It seems the solution of, $$u^2+dv^2 = w^3\tag1$$ involves the class number $h(d)$. Assume $\gcd(u,v)=1$. Q: For which $\color{red}{prime}\; d$ is the complete solution of $(1)$ in the integers ...
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Examples for abstract class field theory?

I'm starting to get into Abstract Class Field Theory, following Neukirch's famous ANT. The initial setup is basically a profinite group $G$ and a discrete abelian group $A$ on which $G$ acting as ...
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Artin Reciprocity $\implies$ Cubic Reciprocity

I'm trying to understand the proof of cubic reciprocity from Artin reciprocity as outlined in this well-known previous math.SE question and the link KCd mentions there. However, there's one final step ...
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Does the prime ideal $(p)$ of $\mathbb{Z}[\sqrt{-5}]$ split completely in the extension of $\mathbb{Q}(\sqrt{-5}, i)/\mathbb{Q}(\sqrt{-5})$?

For prime numbers $p$ such that $p \equiv 11$, $13$, $17$, $19 \text{ mod }20$, does the prime ideal $(p)$ of $\mathbb{Z}[\sqrt{-5}]$ split completely in the extension of $\mathbb{Q}(\sqrt{-5}, i)/\...
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Group cohomology for $\mathbb{Z}[G]$-modules versus $k[G]$-modules.

I am trying to get familiar with group (Tate) cohomology. I am for instance reading Brown's Cohomology of Groups. Now something seems unclear to me What information do we hope to attain from studying ...
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Question on a permutation of the roots of a Galois number field, dihedral over the rationals

The polynomial $f(x) = x^6 - 7x^5 + 21x^4 - 41x^3 + 63x^2 - 63x + 27$ defines a Galois extension $H$ of $\mathbb{Q}$. The Galois group of the extension $H/\mathbb{Q}$ is dihedral, and depending on ...
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stuck on proof of inflation restriction exact sequence in group cohomology

The following proof of the inflation-restriction exact sequence is taken from Milne's notes on class field theory. My question is: why does $\phi':G/H \to M$ actually take values in $M^H$? In other ...
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Importance of the compactness of idele group

Si $k$ is a number field and $J_k$ is the group of $V_k^*$ of invertible elements of the adele ring $V_k$ with the induced topology given by the morphism $V_k\to V_k\times V_K,\ x\mapsto(x,x^{-1})$. ...
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Local reciprocity map applied to norm

This question concerns the local reciprocity homomorphism $r_L : L^\times \to G_L^\text{ab}$, where $L$ is a local field with absolute Galois group $G_L$. If $K\subset L$ is a subfield and $r_K$ is ...
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Compute Takagi group of the extension $\mathbb Q(i,\sqrt{-5})/\mathbb Q(\sqrt{-5})$

Given an extension $L/K$ of number fields we define the Takagi group as the subgroup $$T_{L/K} = N_{L/K} (D_L) \cdot H_K \subseteq D_K$$ where $N_{L/K}$ is the relative norm, $D_\bullet$ is the ...
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Dirichlet density for number fields $K$?

Let $K$ be a number field. Let $P$ be a subset of the set of nonzero prime ideals in $K$. For $\mathfrak{p} \in P$, let $N(\mathfrak{p})$ be its absolute norm, so $N(\mathfrak{p}) = \#\mathcal{O}_K/\...
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Which primes ramify when adjoining roots of a unit?

1) For ANF $K$, if $\zeta_n, u\in\cal O_k^\times$, are there any primes that ramify in $k(\sqrt[n]{u})/k$? 2) is the HCF composed solely of all such extensions, or are there others? Long story: Let $...
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how can I prove this statement in local field theory?

Let $\mathbb{K}$ be a fixed local field. Then there is an integer $q=p^r$, where p is a fixed prime element of $\mathbb{K}$ and r is a positive integer [edit] such that for each $x\in\mathbb{K}, x\...
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Is $\mathbb{Q}(\zeta_5)/\mathbb{Q}(\sqrt5)$ the maximal finite abelian extension of $\mathbb{Q}(\sqrt5)$ unramified away from $5\infty$?

The following problem appears in a homework question posed by B. Conrad (2(i) here: http://math.stanford.edu/~conrad/249BPage/homework/hmwk9.pdf): Using class field theory, prove that $\mathbb{Q}(\...
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Group cohomology or classical approach for class field theory?

First of all, I don't think this is a duplicate, because the related questions I found were mainly about history of group cohomology in number theory and there was no one asking about the classical ...
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Divisibility of Differents in Tower of Fields

In Lemmermeyer's class field theory notes, on page 114 there is the following claim. Let $F = \mathbb Q(\sqrt{-5})$, then let $K = F(\sqrt{-1})$. Let $F_1 = \mathbb Q(\sqrt{-1})$ and $F_2 = \mathbb Q(\...
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Variant of strong approximation.

Let $K$ be a global field. Let $w$ be a place of $K$. Let $\textbf{A}^w$ be the restricted direct product over all $v$ except $w$ of the $K_v$ with respect to the subgroups $\mathcal{O}_v$. How do I ...
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kernel of the artin map when dealing with S-ideles and S-divisors for function fields

Let $L/K$ be an abelian extension of function fields and let $\vartheta_{L/K}:\mathcal{D} \to \text{Gal}(L/K)$ be the Artin map from the divisors of $K$ to the galois group. What can be said about ...
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Non-open subgroups of finite index in the idele class group of a number field

Let $K$ be a number field and $C_K$ be its idele class group. Exercise 5.11 in Milne's notes asks me to show that there is a finite-index subgroup $H$ of $C_K$ which is not open. I haven't found an ...
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Why is the group of principal units of a local field uniquely divisible by $n$?

I am reading a proof with the followings setup and claim. $K/F$ is a Galois extension of local fields with group $G$ of order $n = q^s$, where $q$ is prime and $s \geq 1$. Assume the maximal ideal $\...
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Class field theory for $p$-groups. (IV.6, exercise 3 from Neukirch's ANT.)

I will use notation as in a preious question of mine. This question is from Neukirch's book "Algebraic number theory," page 305, exercise 3. Notation for the problem Let $G$ be a profinite $p$-...
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Neukirch and a congruence condition.

I'm reading Neukirch's book "Class Field Theory" (from 1980, not the newly translated one including cohomology) and there's one statement that I can't wrap my mind around, and I suspect it stems from ...
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A specific problem on Class field theory

Let $K$ be a quadratic complex number field. Let $p$ be a prime greater than $5$ unramified in $K/\mathbb{Q}$. Let $M$ be the compositum of all finite $p$- extensions of $K$ which are unramified ...
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Metric on $\mathbb{Q}$ for which the completion of $\mathbb{Q}$ is isomorphic to $\mathbb{Q}_2 \times \mathbb{Q}_3$. [closed]

Is there a metric on $\mathbb{Q}$ for which the completion of $\mathbb{Q}$ is isomorphic to $\mathbb{Q}_2 \times \mathbb{Q}_3$?
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Not affine, projective, geometrically connected, geometrically reduced, nor geometrically regular…

Is there a field $k$ and a regular integral $k$-variety $X$ that is neither affine, projective, geometrically connected, geometrically reduced, nor geometrically regular?
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Primes of special form

Are there infinitely many primes $p$ of form $$2^k+a^2=p^2<2^{k+2}$$ where $a\in\Bbb N$? Which primes are known to be of such form? An example is $16+3^2=5^2$. This is the only one I could find.
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Sum of squares as Primes Class Field Theorem statements

We know that every prime $1\bmod 4$ can be written in an unique way as $a^2+b^2$ form where $a,b\in\Bbb N$. Is there a comprehensive list of other statements of form "every prime $d\mod r$ can be ...
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Hilbert 2-Class Field definition

what is a Hilbert 2- class field? As a Hilbert Class field of a number field K is the maximal unramified abelian extension, of K,
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Quartic extension

Consider primes represented by $x^2 + ny^2$. If $m$ and $n$ are two positive integers such that the Legendre symbol $\left(\frac{-m}{n}\right) = 1$, then does there exist a cyclic quartic unramified ...
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Primes of the form $x^2+n\cdot y^2$, given $n$?

In an attempt to get to grips with algebra for a course I intend to follow, I was working through a bunch of exercise sheets. A series of questions got me wondering: Given an integer $n$, is there ...
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What are the restrictions in the ramification behavior of a Galois extension of number fields imposed by the Galois group of the extension?

Studying class field theory, I have come across the following Proposition: Proposition. Let $K/E$ be an extension of number fields so that there is no nontrivial unramified subextension $F/E$ with $...
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Ramification in $\mathbb Q(i,\sqrt[4]\pi)/\mathbb Q(i)$

Let $\pi\ne1+i$ be a prime element of $\mathbb Z[i]$. I am interested in the ramification in the extension $\mathbb Q(i,\sqrt[4]\pi)/\mathbb Q(i)$, especially over $(1+i)$. I've tried for instance to ...
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Given $d \equiv 5 \pmod {10}$, prove $\mathcal{O}_{\mathbb{Q}(\sqrt{d})}$ never has unique factorization

With the exception of $d = 5$, which gives $\mathbb{Z}[\phi]$, of course (as was explained to me in another question). I'm not concerned about $d$ negative here, though that might provide a clue I ...
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$\mathbb{F}_p[T, 1/T]$ is discrete in $\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T))$, adeles.

Let $p$ be a prime number. How do I show that $\mathbb{F}_p[T, 1/T]$ is discrete in $\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T))$?
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Reference request: principalization theorem

Let $K$ be a number field, and $\mathbb{I}_K$ the group of ideles. The Hilbert class field $M$ of $K$ is the class field of the open subgroup $H = K^{\ast} \mathbb{I}_K^{S_{\infty}}$, where $$\mathbb{...
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Norm map and extension of idele class groups

It's a classic fact in class field theory that for an extension of number fields $L/K$, we have the norm map on idele class groups $\mathcal{N}:C_L\to C_K$ descended from the map on ideles given by ...
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Serge Lang Never Explains Anything Round II

I'm reading the second edition of Lang, Algebraic Number Theory, page 221. I quote: Let $F$ be a local field, i.e. the completion of a number field at an absolute value. Let $L$ be an abelian ...
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Which properties determine the uniqueness of the global Artin map?

Let $L/K$ be a finite abelian extension of number fields. There is a well defined homomorphism $\Phi: J_K \rightarrow G(L/K)$, called the Artin map, with the following properties (among many): (i) $\...