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

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Fixed fields in Neukirch's book (chap. IV): notational problem

I am reading chapter IV of Neukirch's ANT, and there is a thing that I don't understand. First of all I have to introduce the notations of chapter IV. $G$ is a profinite group and: Clearly this ...
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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}) = ...
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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}, ...
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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 ...
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59 views

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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59 views

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 ...
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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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1answer
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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 ...
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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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1answer
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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 ...
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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) ...
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idelic ray class group modulo $\mathfrak{m}$

I'm studying the idele group $\mathcal{I}$ for a number field $K$. My definition of the ray class group attached to a modulus $\mathfrak{m}$ is $$\mathcal{C}_{\mathfrak{m}}= ...
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Proof of direct sum of ideal class group of Neukirch book

In books Neukirch, Algebraic Number Theory. I don't understand. 1) Why there exists $a$ such that $a\equiv c \ \mod \mathfrak p $ and $a\in ca_{\mathfrak p}^{-1}a_{\mathfrak q}$ for $\mathfrak ...
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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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1answer
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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 $\mathfrak p$ ...