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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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Examples of Lubin-Tate formal groups

I'm currently learning about Lubin-Tate theory: Given a p-adic field $K$ and a uniformizer $\pi \in \mathcal{O}_K$, they consider formal $\mathcal{O}_K$-modules $F$ such that the endomorphism $[\pi]$ ...
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Norm surjective for unramifeid extension of local fields $L/K$

I have a question about the argument used here in proof of Proposition 4.2.5.: For any finite unramified extension $L/K$ of local fields, the map $\text{Norm}_{L/K}: O_L^* \to O_K^*$ is surjective. ...
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Are Hecke L-functions associated to Artin L-functions primitive?

I'm reading a famous paper by Lagarias & Odlyzko on effective versions of the Chebotarev density theorem. There is one thing about Artin L-functions that is a little perplexing to me. Let $L/K$ be ...
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Is there any relationship between the study of class number of a field and the study of class field theory through Lubin-Tate formal group?

I am curious to know if one can relate to the study of Lubin-Tate formal group and related local class field theory with the study of class number of a field (global field in general). As far as I ...
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If $\frak{a}$ is a proper ideal of $\mathcal{O}_K$, then $\bigcap_{i\geq 1} \frak{a}$ $^i =\{0 \}$.

I am reading these notes on the Kronecker-Weber theorem. On page 9, they quote without proof the theorem: If $\frak{a}$ is a proper ideal of $\mathcal{O}_K$, then $\bigcap_{i\geq 1} \frak{a}$$^i =\{0 ...
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Understanding a key definition in Lagarias Odlyzko's paper on Chebotarev density theorem

Lagarias & Odlyzko has a 1977 paper where they prove effective versions of the Chebotarev density theorem. I am having trouble understanding equation (3.1). Here, $L/K$ is a Galois extension of ...
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Norm $\text{N}_{L/ \Bbb Q_p}(\pi_L)$ of Uniformizer of finite extension of $p$-adics

Let $ K/ \Bbb Q_p$ a finite extension of $p$-adic field $\Bbb Q_p$ of degree $[K:\Bbb Q_p]=n$ and let $ \pi_L$ be a uniformizer of $L$. Question: Can we say something interesting / "distinguished&...
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Upper estimation for Multiplicative Conductor of Extension of $\Bbb Q_p$

Let $L/\Bbb Q_p$ be a finite extension of $p$-adics of degree $n$. Let $f$ be the minimal number such $1 +(p)^f:=U^f \subset \text{Norm}_{L/ \Bbb Q_p}(L^*)$. The ideal $(p)^f$ is also called ...
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Reconciling different ideal-theoretic definitions of Hecke Characters

I'm reading Chapter 7 of Neukirch's book on algebraic number theory, where the author defines a Größencharakter (6.1) as: Let $\mathfrak{m}$ be an integral ideal of the number field $K$, and let $J^{\...
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When can we divide a shape into n equal parts (like Gauss and Abel)?

Gauss' theorem on the constructibility of the regular n-gon is very famous. I recently came across the similar theorem by Abel on dividing the lemniscate into n equal parts, and wondered if there are ...
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Computation of Norm

I am attempting an exercise from a Galois Theory problem sheet I found online. It asks: Let $p$ be an odd prime. Let $L_1 = \mathbb{Q}_p(\zeta_p) / \mathbb{Q}_p$ and let $L_2=\mathbb{Q}_p(\sqrt[p-1]{-...
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Artin map is the Kronecker symbol

Let $d$ be a squarefree integer. Then the discriminant $\Delta$ of $\mathbb{Q}(\sqrt{d})$ over $\mathbb{Q}$ is either $d$ or $4d$. I know that a prime number $p$ is ramified in $\mathbb{Q}(\sqrt{d})$ ...
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Splitting of primes in abelian extensions

Let $K/\mathbb{Q}$ an abelian number field. Using the Kronecker-Weber theorem, it is quite easy to show the following statement: There is a number $m\in\mathbb{N}$ and a finite set $M\subseteq \{0,1,.....
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How to calculate the conductor of Dirichlet character associated to a Galois representation?

I am reading Serre's 1972 paper - in section 5.6, he considers an $\ell$-adic representation $\rho_{\ell}(G)$ to lie in a Borel subgroup (i.e., upper triangular matrices). He then says there ought to ...
Yang Awotwi's user avatar
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Working with the norm map of ideles

Firstly, let me define the relevant notation. Let $L/K$ be an abelian extension of number fields. We let $\mathbb{I}_K$ be the ideles of $K$, $C_K = \mathbb{I}_K / K^\times$ the idele class group, $\...
Daniel New's user avatar
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Kernel of the idele to ideal map $( \cdot ): \mathbb{I}_K \to I_K$ with restricted domain and codomain.

I'm a little confused about this step that is made in a proof I'm attempting to understand. For some notation, we have $\mathbb{I}_K$ to be the ideles of $K$, $I_K$ to be the fractional ideals of $K$, ...
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References on Narrow Ring Class Fields

I am in the process of writing my master's thesis about the 2018 article of Darmon and Vonk about a Nonarchimedean approach to a real multiplication theory (https://www.math.mcgill.ca/darmon/pub/...
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What are the cohomology groups $H^n(k, \mathbb{Q/Z})$ for a field $k$?

I'm reading a source that connects Brauer groups of fields $k$ to $H^1(k, \mathbb{Q/Z})$, but the source doesn't define the cohomology groups $H^\bullet(k, \mathbb{Q/Z})$. Googling the definition ...
mattematician 's user avatar
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Cohomology of roots of unity vs cohomology of $\mathbb{Z}/n\mathbb{Z}$

Let $K$ be a local field of chracteristic zero and let $n \in \mathbb{N}$. I want to see whether the cohomology groups $H^1(Gal(\bar{K}/K); \mu_n)$ and $H^1(Gal(\bar{K}/K); \mathbb{Z}/n\mathbb{Z})$ ...
coconuthead's user avatar
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Why is $K \cap \mathbb{Q}^{\text{cyc}}=\mathbb{Q}$ iff $\chi_K(G_K)=\hat{\mathbb{Z}}^{\times}$?

I am reading David Zywina's "Elliptic curves with maximal Galois action". For a number field $K$, he defines $\mathbb{Q}^{\text{cyc}} \subset \overline{K}$ to be "the" cyclotomic ...
Batrachotoxin's user avatar
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irreducibility in p-adic field

Let $u\in \mathbb{Z}_p^*$ be a unit in the ring of $p$-adic integers. Assume that $u^{1/p}\not\in \mathbb{Q}_p$, in other words $u$ is not a $p$-power. I am wondering how the polynomial $f=x^p-u$ ...
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local l-th power at some place

Suppose that $F$ is a global field containing $l$-th roots of unity where $l$ is a prime integer and $l\neq\mbox{char}\ F$ and that $S$ is a finite set of places $v.$ Given some $t_v\in F_v(c^{1/l})$ ...
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quotient group of Idele group is its own Pontryagin dual with respect this pairing

In the J. Tate's paper "Relations Between K2 and Galois Cohomology" Lemma 5.2, it says that given $\alpha_1,\cdots,\alpha_r\in $Br$_lF$, i.e. $\alpha_i$ is killed by $l$, where $l$ is prime ...
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The Herbrand quotient of a valuation ring

My question stems from trying to solve the following problem: My goal is to calculate the Herbrand quotient of $\mathcal{O}_L^\times$ by hand, so only using elementary calculations in $\mathbb{Q}_p$. ...
coconuthead's user avatar
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ch. 8.3 Exercise 1 in Cohomology of Number Fields

Let $k$ be a number field, $S$ a set of places of $k$, $k_S$ the maximal extension of $k$ unramified outside $S$, $\mathcal{O}_S$ the subring of $k_S$ with $\nu_{\mathfrak{p}}(\alpha)\geq 0$ for all $\...
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Serre's definition of $U_{\mathfrak{m}}$ for $\ell$-adic representations

In Serre's "Abelian $\ell$-adic representations and Elliptic curves", in order to define the set $S_{\mathfrak{m}}$ he defines of $U_{v,\mathfrak{m}}$ as follows: $$U_{v,\mathfrak{m}}=\left\{...
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Ramification of ray class field/ ring class field

Suppose $K$ is a number field. Let $q$ be a prime of $K$. Let $ K [q] $ be the ray class field of $K$ of modulus $q$. Someone claims that $K[q]$ is totally ramified at $q$ over the hilbert class field ...
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Galois group of the maximal extention unramified outside a finite set of primes over a number field

In Karl Rubin's book Euler System, he gives a definition of selmer groups as below: Then he states as in lemma 5.3 on page 12 that the selmer group with upper tag is equal to a first cohomology group ...
Taozipeter's user avatar
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Understanding the isomorphism $\widehat{K^*} \cong {\mathcal{O}^*_K} \times \widehat{\mathbb{Z}}$

Let $K$ be a local field and let $\widehat{K^*}$ and $\widehat{\mathbb{Z}}$ denote the profinite completions of $K$ and $\mathbb{Z}$. As the title suggests I'm having difficulties understanding the ...
coconuthead's user avatar
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There exists imaginary quadratic extension which trivialized 2-part of ideal class group

Let $p$ be a negative prime number such that $p \equiv 5\pmod 8$. Let $K = \mathbb{Q}(\sqrt{p})$ and denote its ideal class group by $Cl_K$. I aim to prove that $Cl_K[2] := \{a \in Cl_K \mid 2a = 0\}$ ...
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Maximal abelian unramified outside S extension of exponent m of K where S is a finite set of places of K is finite

I saw a rather sleek proof of the following fact: Let $K$ be a number field. Let $L/K$ be a maximal abelian unramified outside $S$ extension of exponent $m$ of $K$ where $S$ is a finite set of places ...
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Example on Childress; Quadratic Reciprocity

In Childress'Class Field Theory book, section 5.3 is an example of how to deduce the Classical Quadratic Reciprocity Law from the idelic version of Artin's Reciprocity Law. It begins by constructing ...
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associate a character to an abelian extension in local class field theory

Let $K_v$ a local and $F$ a finite abelian extension of it. Two questions: Could somebody explain how local class field theory associates naturally a character of $K_v^{\times}$ to $F$. And, if $r \in ...
JackYo's user avatar
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class number of quadratic field with discriminant $17$ [duplicate]

I'm trying to prove that class number of the quadratic field with discriminant $17$ is $1$. Let $K =Q[\sqrt{17}]$ be the quadratic field and $\mathfrak{o}_{K}=\mathbb{Z}[\frac{1+\sqrt{17}}{2}]$ be the ...
unoriginalname's user avatar
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Show by Local Class Field Theory $\mathbf Q_p$ has unique Galois ext. iso. to $(Z/2Z)^2$ if $p > 2$, and unique Galois ext. iso. to $(Z/2Z)^3$ o/w.

Show that $\mathbf{Q}_p$ has a unique Galois extension isomorphic to $(Z/2Z)^2$ if $p > 2$, and that $\mathbf{Q}_2$ has a unique Galois extension isomorphic to $(Z/2Z)^3$ I have already completed ...
Rick's user avatar
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1 answer
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Prove that for a imaginary quadratic field $K/\mathbb Q$, $Cl(K)/2Cl(K)= (\mathbb Z/2)^{r-1}$ where $r$ is the number of prime that ramify in $K$.

I want to prove that for a imaginary quadratic field $K/\mathbb Q$, $Cl(K)/2Cl(K)\simeq (\mathbb Z/2)^{r-1}$ where $r$ is the number of prime that ramify in $K$. This question has appeared on this ...
mathemather's user avatar
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Find an abelian extension of 2-adic rationals such that $Gal(K/\mathbb Q_2)= \mathbb Z/2 \times (\mathbb Z /2^r)^2$.

Let $r>0$. I want to prove that there exists a extension $K / \mathbb Q_2$ such that $Gal(K/\mathbb Q_2)= \mathbb Z/2 \times (\mathbb Z /2^r)^2$. This is an exercise in Kedlaya's notes on class ...
mathemather's user avatar
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Lang exercise 50 on Witt vectors

I'm reading the construction of the Witt ring from Lang's algebra. This is a series of exercises in chap. VI. In exercise 50 he says: If $x\in W_n(k)$ show that there exists $\xi\in W_n(\bar{k})$ ...
Mea Culpa's user avatar
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Hasse Symbol for ramified primes

My Reference is https://virtualmath1.stanford.edu/~conrad/249BW09Page/handouts/cfthistory.pdf In chapter 7, page 17, Keith says For example, $(\alpha, L/K)_v$ lies in the common decomposition group $...
Assaf Marzan's user avatar
4 votes
2 answers
474 views

How important is getting nitty-gritty with ideals for algebraic number theory?

Coming off an undergraduate course on number fields based on Marcus's textbook Number Fields, I am interested in taking the logical next step towards (local) class field theory, as well as Iwasawa ...
Jake Lai's user avatar
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What is the local character given by class field theory?

Dorman's paper on singular moduli uses the "local character given by class field theory". Specifically, on page 178, the author states, For each prime $p$, finite or infinite, let $\...
stillconfused's user avatar
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"Classical" approach to local class field theory (Brauer group and Hasse invariant)

I'm trying to learn local class field theory from the corresponding chapter of Lorenz's "Algebra 2", which, according to its mission statement, takes a more classical approach [Local class ...
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Normal extension of given degree of a number field

Let $n>1$ be a natural number. Question. Let $K$ be a number field, and let $S\subset V_f(K)$ be a finite set of finite (non-archimedean) places of $K$. Does there exist a normal extension $L/K$ ...
Mikhail Borovoi's user avatar
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154 views

Maximal p-extension and pro-p extension

I’m studying Iwasawa theory and I meet some questions Thanks a lot for your help. Q_1: About terminology $p$-extension. I find many reference use maximal $p$-extension or maximal abelian p-extension ...
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Formal groups over discrete valuation rings

Let $k/\mathbb{Q}_p$ be a finite extension. Let $k^{ur}$ be its maximal unratified extension and consider $K$ its completion. Let $\varphi$ be the Frobenius morphism in Gal$(k^{ur}/k)$ and whit the ...
Mario's user avatar
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Finding class group of $\Bbb Q(\zeta_{29})$

I wondered the class group $G_{29}$ of the cyclotomotic field $\Bbb Q(\zeta_{29})$ today (29 October) which is the second non-trivial class group of the sequence of fields $\Bbb Q(\zeta_p)$ where$\...
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A localization sequence for étale cohomology of $\mathcal O_K$, where $K$ is a local field

Given a non-archimedean local field $K$, let $\mathcal O_K$ be the associated valuation ring and $k$ its residue field. According to this MO answer, we have short exact sequence $$0 \to H^2(\mathcal ...
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Torsion-free modules over an order

Let $F/\mathbb Q$ be a field extension, let $\mathfrak o$ be an order in $F$ (this is, $\mathfrak o$ is a subring which is free as a $\mathbb Z$-module, of rank $[F:\mathbb Q]$). Are there any results ...
Aitor Iribar Lopez's user avatar
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Ramification of primes in ring class fields

Let $K$ be an imaginary quadratic field of discriminant $D_K$, let $n$ be a positive integer. Call $H_n$ the ring class field of $K$ of conductor $n$. Then, all primes of $K$ that do not divide $n$ ...
Fraz's user avatar
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1 answer
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How to determine the minimal polynomial of the higher power of the Weber f function?

There is a certain theorem that guarantees that the function value of the higher power of the Weber f function is an algebraic integer \begin{align*} x\stackrel{?}{=}\mathfrak{f}^8(\tau)=\left(\frac{\...
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