The class-field-theory tag has no usage guidance.

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### A cyclic subgroup as a decomposition group

Let $G$ a finite group appeared as galois group of an extension of $\mathbb{Q}$. Is it true that any cyclic subgroup $C \subset G$ can be realized as a decomposition group of an ideal $\mathfrak{P}$ ...

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96 views

### The Galois side of the Norm map

Let $K$ be an abelian extension of $\mathbb{Q}$. We know that $[x, K]|_{\mathrm{Gal}{\mathbb{Q}^{ab}}}=[\mathrm{N}^{K}_{\mathbb{Q}} x, \mathbb{Q}]$ where $[x, F]$ is the Artin reciprocity map. Given a ...

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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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### A prime ideal $\mathfrak{p}$ decomposes in $\mathbb{Q}(\zeta_{24})/\mathbb{Q}(\sqrt{-6})$ iff it is generated by $\alpha\in1+2\Bbb{Z}[\sqrt{-6}]$

For a nonzero prime ideal $\mathfrak{p}$ of $\mathbb{Z}[\sqrt{-6}]$ which does not divide $2$, does $\mathfrak{p}$ decompose completely in the extension $\mathbb{Q}(\zeta_{24})/\mathbb{Q}(\sqrt{-6})$ ...

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467 views

### Artin reciprocity $\implies $ Cubic reciprocity

I asked this on math.SE a few days ago with no reply, so I'm reposting it here. Hope this is not considered too elementary for MO (feel free to close if so).
I'm trying to understand the proof of ...

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### extending $p$-adic character of the local intertia to the absolute Galois group

Suppose I have a number field $F$, and a finite place $v$ of $F$. Let $E$ be finite extension of $F_v$. I start with a continuous morphism
$$
\chi \colon O_{F_v}^\times \to E^\times.
$$
where ...

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325 views

### Are the abelian absolute Galois groups of these local fields isomorphic?

For a field $F$ we denote by $F^{\mathrm{ab}}$ the compositum of all finite Galois abelian extensions of $F$.
Is $\mathrm{Gal}(\mathbb{Q}_2(\sqrt[8]{3})^{\mathrm{ab}}/\mathbb{Q}_2(\sqrt[8]{3})) ...

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293 views

### Parametrizing all cyclic extensions of the rational numbers of degree 5

Is there a polynomial $f(T,X) \in \mathbb{Q}(T)[X]$ in the indeterminate $X$ over the field $\mathbb{Q}(T)$ with $\mathrm{Gal}(f/\mathbb{Q}(T)) \cong \mathbb{Z}/5\mathbb{Z}$ such that for every Galois ...

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68 views

### Do character sheaves split over the Lang isogeny?

Let $G$ be a smooth commutative connected algebraic group over a finite field $\mathbb{F}_q$. For my purposes a character sheaf on G is a rank one $\ell$-adic local system $\mathcal{L}$ on $G$ ...

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### Iwasawa theory, $\mathbb{Z}_p^{2}$-extension, Greenberg module

Take $H\subset \bar{\mathbb{Q}}$ be a quartic imaginary number field such that $\operatorname{Gal}(H/\mathbb{Q})=\mathbb{Z}_2 \times \mathbb{Z}_2$. Denote by $F$ the quadratic real subfield of $H$ and ...

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### Cubic modular equations solutions when decomposition field is not a HCF

I was interested in counting (and more generally having somehow an interesting expression) the numbers of solution of cubic equations modulo a prime $p$.
So here are my thoughts.
Let take a cubic ...

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392 views

### What are “Artin fractions”?

The German Wikipedia entry for Ernst Witt https://de.wikipedia.org/wiki/Ernst_Witt has a photo of his grave in Hamburg. The bottom part has a visible text "Artin Brueche" (Artin fractions) but the ...

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### Unramified extensions of quadratic fields

Let $K/\mathbb{Q}$ be quadratic and let $L/K$ be an (everywhere) unramified Galois extension. If $L/K$ is abelian, then one can show that $L/\mathbb{Q}$ is Galois (eg see here). Is $L/\mathbb{Q}$ ...

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### Newly defined $L$-function in terms of $L$-function, does it have any obvious zeros or poles?

Let $K$ be a number field, $Cl(K)$ the ideal class group, $\chi: Cl(K) \to \mathbb{C}^\times$ a homomorphism. If $\mathfrak{a} \subset \mathcal{O}_K$ is any ideal, let $[\mathfrak{a}]$ denote its ...

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### Is the localization of the maximal abelian extension still a maximal abelian extension?

Let $K$ be a number field and consider the maximal abelian extension $K^{ab}$ of $K.$ For a finite prime $p,$ letting $K_p$ be the completion of $K$ at $p,$ we have an extension $K_p \subset K_p ...

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47 views

### Kernel of the Artin map when dealing with S-ideles and S-divisors for function fields

Having understood that there is a strong correspondence between number fields and function fields, I am trying to work out some function field equivalents of class field theoretic invariants from Bost ...

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118 views

### Class field theory for $p$-groups.

I accidentally posted this question to math.stackexchange but think that it is more appropriate here (if not, please say so!):
This question is from Neukirch's book "Algebraic number theory," page ...

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87 views

### On a theorem of Dwork and totally ramified extensions

Suppose that $K \subset L$ is a totally abelian ramified extension of local fields. Let $\pi_L$ be a prime element of $L^*.$ $F \in Gal(\tilde{L}/L)$ is the Frobenius, where $\tilde{L}$ is the maximal ...

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### Motivating Lubin-Tate theory

The Lubin-Tate theory gives an amazingly clean and streamlined way of constructing the subfield (usually denoted) $F_\pi\subset F^\mathrm{ab}$ for a local field $F$ fixed by the Artin map associated ...

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188 views

### What is the relationship between the conductor of an order and the conductor of a number field extension?

What is the relationship between the conductor $\mathfrak{f}_{\mathcal{o}}$ of an order $\mathcal{o}\subset \mathcal{O}_K$ and the conductor $\mathfrak{f}_{L/K}$ of a field extension in the classical ...

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### comparison of completion and Henselization in class field theory

Given a ring $R$ with maximal ideal $\mathfrak{m}$, we can form the localization $R_\mathfrak{m}$, the completion $\hat{R}_\mathfrak{m}$ or the Henselization $\hat{R}^h_\mathfrak{m}$ of $R$ with ...

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### For $K/E$ a number fields extension and $F/E$ a finite Galois extension, how is the ramification in $F\cdot K/K$ related to the one in $F/E$?

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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### Meromorphic functions on $U^2 = T^3 + 1$, cokernel of $O_S \to F_\infty/O_\infty$ [closed]

See here. Crossposted from math.stackexchange since there's no good answer despite $>$ 20 upvotes.
Let $k$ be a field of characteristic $\neq 2$, and consider the quadratic extension $F$ of ...

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### Finite-index subgroups of the ideles

Let $k$ be a number field and denote by $J_k$ the idele group of $k$. Recall that the finite-index open subgroups of $J_k$ which contain $k^*$ are very important in class field theory. My question ...

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### Computation of Hilbert symbol of order 4

We have explicit expressions for the quadratic Hilbert symbol over $\mathbb Q$, for example $\left(\dfrac{x,y}2\right)_2=(-1)^{\frac{x-1}2\frac{y-1}2} (x,y\ne2)$. Are similar expressions known for ...

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### $F[[T]] \times F[[1/T]]$ fundamental domain, show compactness

Let $p$ be a prime number. What is the easiest way to see that $(\mathbb{F}_p((T)) \times \mathbb{F}_p((1/T)))/\mathbb{F}_p[T, 1/T]$ is compact? Here $\mathbb{F}_p[T, 1/T]$ is embedded in ...

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### Can we prove the uniqueness of the local Artin map by using mostly global class field theory?

Let $l/k$ be a finite abelian extension of $p$-adic fields. There is a well defined local Artin map $k^{\ast} \rightarrow Gal(l/k)$ with kernel $N_{l/k}(l^{\ast})$. Let's suppose that we have only ...

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### The use of Haar measure in the 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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### Furtwangler's Principal ideal theorem in number fields

Does anyone know a simple proof, using cohomological method of the fact that the verlagerung from a finite group G. to its commutator subgroup G', i.e. $$G/G'->(G')^{ab}$$ vanishes?
The simplest ...

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### $K^{ur}K^{\pi} = L$

Let $K$ be a $p$-adic field, and $L$ an infinite abelian extension of $K$ containing $K^{ur}$. Let $\Phi: K^{\ast} \rightarrow Gal(L/K)$ be the local Artin map. Let $\pi$ be a uniformizer for $K$, ...

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### Is $K^{ur} K^{\pi} = L$?

Let $L/K$ be a finite extension of $p$-adic fields, $\pi$ a uniformizer of $K$, $\theta = (-, L/K)$ the local Artin map $K^{\ast} \rightarrow Gal(L/K)$, $E$ be maximal unramified extension of $K$ ...

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### Minimal Discriminants

Let $D_n$ be the minimal absolute value of the discriminants of
number fields with degree $n$. Arnold Scholz conjectured in 1936 that
$D_{397} > D_{400}$, which is, of course, still open (Scholz ...

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### Explicit extensions for Heisenberg groups

Let $G$ be the $p$-adic Heisenberg group $\begin{pmatrix} 1&\mathbb Z_p&\mathbb Z_p\\&1&\mathbb Z_p\\&&1\end{pmatrix}$. Is it possible to write an explicit extension $K/k$, ...

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### Quickest and/or most elementary proof of “principal iff splits completely”?

Let $L$ be the Hilbert class field of a number field $K$, and let $\mathfrak{p}$ be a prime ideal of $K$. Then $\mathfrak{p}$ splits completely in $L$ if and only if $\mathfrak{p}$ is a principal ...

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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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### 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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### What are the current trends in class field theory?

Being far from an expert in the subject I was wondering if people can hint towards a modern exposition of the developments in the last 10 years ? Or if not then suggest some sub-subjects in CFT that ...

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### Class field towers

It is known (Golod and Shafarevich) that the class field tower of a finite extension $K$ of $\mathbb{Q}$ may be infinite. But is it always finite for $K=\mathbb{Q}[\zeta]$ where $\zeta$ is a root of ...

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368 views

### Deciding a quadratic diophantine equation

Given $a,b\in\Bbb Q_+$, is there an easy way to decide if $$S_{a,b}=\{(x,y)\in\Bbb Z^2:ax^2 + by^2=1\}=\emptyset?$$
I am more interested in seeing if there is a quick way to test for case when ...

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125 views

### Projective coordinates over a non UFD ring

Is it true that when the integers of a number field are not a UFD then not every point in projective $n$-space over that field can be given by relatively prime algebraic integer coordinates?
When a ...

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### Density of primes of degree one in Bauer's Theorem (Application of Chebotarev Density)

Let $L$ be a Galois extension of $\mathbb{Q}$ and $M$ a finite extension of $\mathbb{Q}$, both of degrees $> 1$. A Theorem of Bauer tells that $Spl_1(M)\subset Spl(L)$ up to a finite number of ...

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### Does the fundamental group identify group structure on subvarieties of products of curves?

Let $C_1,\dots, C_n$ be smooth curves over $\overline{\mathbb F}_p$, not necessarily proper. Let $X$ be a subvariety of $C_1 \times \dots \times C_n$. I'm interested in the natural map:
$$ ...

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### Unramified extensions of a given degree

Let $K \neq \mathbb{Q} $ be a finite extension of $\mathbb{Q}$. For a given integer $n$, how to construct an unramified extension of $K$ of degree $n$ ?
EDIT: If not then under what conditions on ...

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### Group laws in class field theory

In the case of a quadratic imaginary number field one can construct its maximal abelian extension using torsion points of an elliptic curve with complex multiplication by this field.
In the case of a ...

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### Finite Nontrivial Unramified Towers of Number Fields

Let $F$ be a number field and $L=F^{un}$ its maximal unramified extension. By Class Field Theory, $$Gal(L/F)^{ab}\cong Cl(F).$$ It's well-known that we can have $[L:F]=1$ (e.g. $F=\mathbb{Q}$), and ...

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### Explicit description/calculation of norm group of ideles of characteristic $p$ global field

I posted the same question earlier in stack exchange,
(http://math.stackexchange.com/questions/1130391/algebraic-proof-of-2nd-inequality-of-global-class-field)
thinking it is most definitely not a ...

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### Diophantine equations over cyclotomic fields

Let $\mathbb{Q}^{\text{ab}}$ be the compositum of all finite abelian extensions of $\mathbb{Q}$. Explicitly, $\mathbb{Q}^{\text{ab}}$ is the field obtained from $\mathbb{Q}$ by adjoining all roots of ...

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### A certain idele class character

Let $E/K$ be a cubic extension of number fields, $\nu$ be a Grossencharacter of the idele class group $\mathbb{I}_{E}/E^{\ast}$ such that $\nu^2$ is trivial and $\nu$ restricted to the idele class ...

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### Positive binary quadratic form plus univariate monic cubic (giving Hilbert class field)

We have the Lucas numbers, $$ L_1 = 1, \; L_2 = 3, \; L_3 =4, \; L_4 = 7, L_5 = 11, \; L_{n+2} = L_{n+1}+ L_n \; . $$
Question: is it the case that
$$ f(x,y,z) = 4 x^2 + 3 x y + 9 y^2 + z^3 + 3 z ...