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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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How to prove $(x^{-1})^{-1}=x$ with $\mathbb{K}\backslash\{0\}$ with field axioms? [on hold]

can someone give me a hint, like a base... I wanted to start with the following equation: $x\cdot x^{-1}=1$ but came to no process. Any ideas?
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Proof of Chebotarev Density Theorem without Artin Reciprocity

I'm wondering if there is a proof of the Chebotarev density theorem that does not require the use of any big results in class field theory, such as Artin Reciprocity. As I understand it, the main ...
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Chebotarev's theorem on cyclotomic extensions.

Let $K=\mathbb{Q}$ and $L=\mathbb{Q(\zeta_n)}$, where $\zeta_n$ is the $n$-th primitive power of unity. If $C$ is a conjugacy class of $G=\text{Gal}(L/K)$, Chebotarev's density theorem says that the ...
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What is the discriminant of this order $A=\mathbb{Z}[\alpha]\cap\mathbb{Z}[\alpha^{-1}]$?

Let $f(x) = a_dx^d + a_{d-1}x^{d-1} + \ldots + a_1x + a_0$ be a non-monic irreducible polynomial in $\mathbb{Z}[x]$ and call $\alpha$ be one of it's roots. Let $\mathbb{K}=\mathbb{Q}[\alpha]$ the ...
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Valuation of a number field element over a prime ideal in an order

Having read https://mathoverflow.net/questions/144671/number-field-sieve-for-factorization-with-non-monic-non-linear-polynomial-cant I stumbled on a problem I can't prove. Most of the questions posed ...
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(Milne CFT p. 36) Showing an $A$-module in an exact sequence is cyclic.

I am currently working on the following proof from Milne's Class Field Theory. Here $K$ is a nonarchimedean local field, $A$ its valuation ring $A = \{\alpha \in K: | \alpha| \leq 1\}$. So $A$ is a ...
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Conductor of cyclic cubic extension of number field

I'm currently reading Datskovsky and Wright's "Density of discriminants of cubic extensions" and came across the following (paraphrased) statement. Let $K$ be a number field, and $K'$ a cyclic cubic ...
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What's the relation between class group and class field theory?

It's clear for me what is the relationship and difference between a group and a field in abstract algebra. But, I just started with algebraic number theory and it is not clear to be how (ideal) class ...
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Abelian extensions and the Artin map

I'm trying to understand a proof of the following: Let $L$ and $M$ be Abelian extensions of $K$ (a number field). Then $L\subset M$ if and only if there is a modulus $\mathfrak{m}$, divisible ...
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How many equivalence classes of squares are there?

Let's say we have a $3×3$ square where $3$ of the cells are labeled $a$, $b$, $c$ and the rest are blank. Two such squares are considered "equivalent" if one square can be obtained from another square ...
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Generalized Principal ideal theorem for ray class groups

I 've seen the following statement in https://mathoverflow.net/questions/94480/generalization-of-hilbert-94-and-capitulation and https://mathoverflow.net/questions/63465/where-does-the-principal-...
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Meaning of Hasse-Arf theorem

I am reading about the Hasse-Arf theorem in Serre's 'Local Fields' and I have a hard time understanding what exactly it means for the upper numbering to have jumps only at integers. It seems like a ...
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Brauer Group for a Global Field with $l$-roots of unity $l\neq \text{char}(F)$

Let $F$ be global field that contains the $l$-roots of unity with $l$ a prime number different with the characteristic of $F$ and $\text{Br}F$ the Brauer Group of $F$. How can i proof that all ...
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Why is $\mathbb{C}$ over $\mathbb{R}$ considered ramified?

For a number field $K/\mathbb{Q}$, we say that a finite place of $Q$ is ramified if there exists a valuation $v_{p_i}$ in $K$ lying over $v_p$ such that it is ramified in the sense of the associated ...
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Class field axiom III-1 in Serre's “Local Fields”

Let $F/E$ be a finite extension of local fields, and $N = N_{F/E} : F^* \to E^*$ be the norm map (it is continuos). And consider an axiom: $N$ has closed image and compact kernel. In the book, the ...
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cyclic extension of prime power of a local field

Let $K$ be a non archimedian local field of characteristic $p>0$ with residue field $\mathbb{F}_p$ and $l\neq p$ be a prime. It is known by local classfieldtheory that any abelian Galois ...
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$n\in \mathbf{N}$ such that a solution of $X^4+nX^2 +1$ is a root of unit

Consider $f_n(X)=X^4+nX^2 +1$ in $\mathbf{Q}[X]$. I found that for all natural $n$ such that $n\neq 2-m^2$ for a natural $m$, $f_n(X)$ is irreducible in $\mathbf{Q}$. Consider $K_n=\mathbf{Q}(x)= \...
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Artin reciprocity

The Artin reciprocity says that if $L/\mathbb{Q}$ is a finite abelian extension with defining modulus $m$, then the sequence of groups $$ 1\to I_{L,m}\to (\mathbb{Z}/m\mathbb{Z})^\times\to Gal(L/\...
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About the Artin symbol

Let $L$ be a finite abelian extension of $\mathbb{Q}$ and let $m$ be a positive integer such that $L\subset\mathbb{Q}(\zeta)$, where $\zeta$ is a primitive $m$-th root of unity. Let $a$ be an integer ...
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Totally Tamely Ramified Field Extension

I have a question about @nguyen quang do's answer given in following thread: correspondence between totally, tamely ramified extension and value groups We consider a totally tamely ramified field ...
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Field Extension $k(t^{1/n})/k(t)$ Cyclic

Let $k$ be a field. We consider the field extension $k(t^{1/n})/k(t)$. My question is why and how to see that it is cyclic, therefore $Gal(k(t^{1/n})/k(t))$ is a cyclic group. There is one case that ...
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Understanding problem in Milne's notes on class field theory

I was going through Milne's notes on class field theory and approached the following difficulty in understanding: On page 155, after the discussion of the conductor, he just writes that any Abelian ...
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1answer
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Upper bound for class number of field

Suppose $K$ is a number field. Let $p$ be the smallest prime which is the norm of some principal ideal. It follows that every class in the ideal class group of $K$ contains some ideal of norm prime $q&...
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p-th powers in p-adic field

Denote by $K$ the completion of $\bigcup_{n \geq 1} \mathbb{Q}_p (\zeta_{p^n})$, where $\zeta_{p^n}$ is a $p^n$-th root of unity. Is it true that any element in $K$ is a $p$-th power of some element ...
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1answer
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Is a “local enough” class number always equal to one? [duplicate]

Let $F$ be a number field, and $\mathcal{O}$ its ring of integers. Is there always a finite set $S$ of places of $F$ such that $\mathcal{O}_S$ has class number one? Is it a consequence of standard ...
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What does $(a,b)_{\zeta}$ correspond to in $\mathrm{Br}(\mathbb{Q}_p)=\mathbb{Q}/\mathbb{Z}$

Let $p$ be a prime number, let $\mathbb{Q}_p$ be the local field, by Hensel's lemma, we know it has $p-1$-th roots of unity, let $\zeta$ be a fixed primitive $p-1$-th root of unity in $\mathbb{Q}_p$. ...
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1answer
70 views

Class field theory(Generalization to infinite extension)-Neukirch(General reciprocity law)

My Question ist regarding an exercise 4 in Neukirch Chapter IV § 6 General reciprocity law page 305. Let $G$ be a profinite group and $A$ a $G$-module. Let $(d:G\to \hat{\mathbb Z}, v:A_k\to \hat{\...
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If $p\equiv1\pmod{4}$, do we have $\sqrt{p}\in\mathbb{Q}(\zeta_p)$?

Why is it true that if $p\equiv1\pmod{4}$ then $\sqrt{p}\in\mathbb{Q}(\zeta_p)$?
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Why is $F(u^{1/e})/F$ unramified?

I'm reading a proof of Kronecker-Weber theorem, but I don't know a step in the proof of the below proposition: Prop. Let p be a prime number, $K/\mathbb Q_p$ be a cyclic extension of degree $l^r$ ...
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1answer
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Writting Legendre Symbol as an element of group cohomology of $\mathbb{Q}$

Is it possible to write the Legendre symbol as an element of the cohomology of some kind? We certainly have that it is multiplicative in both numerator and denominator: $$ \left( \frac{a}{p} \right)\...
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The local invariant map is a group homomorphism

Let $ (K, \nu) $ be a nonarchimedian local field. I have read that the Brauer group, $ \text{Br}(K) $ (which for me, is defined by the similarity classes of CSAs with group operation as tensor product)...
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1answer
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Inclusion between norm groups in the idèle class group

Reading Cassels and Fröhlich Chap. VII about Global Class Field Theory, I stumbled upon the following problem: if $K\subset L\subset M$ are finite abelian extensions, then the Main Theorem on Abelian ...
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Examples of unramified abelian extensions of $\mathbb{Q}[i]$

Let me ask a few simple concrete questions (whose answers I’m sure are well known) to motivate my study of class field theory: What is the maximal abelian unramified extension of $\mathbb{Q}[i]$? (I ...
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Reference book/ Notes/ articles for p rank

I am a graduate student. I want to learn about $p$-rank and $p$-torsion points on curves over finite fields. As the base, I have read Stictenoth's book ''Algebraic Function Fields and Codes'' but he ...
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Neukirch ANT - proving reciprocity map is multiplicative

I am learning class field theory from the famous ANT book by Neukirch, where I stuck at the middle of a long proof, whose goal is to prove multiplicativity of reciprocity map. Defining various ...
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Galois Group as Semi-Direct Product

I am trying to understand the following argument but probably have some stupid misunderstanding. Let $ K = \mathbb{Q} ( \sqrt{-D} ) $ be an imaginary quadratic field, and let $ K_{n} $ be the ring ...
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Why $\mathbb Q_p(\xi_n) = \mathbb Q_p$ implies $n|(p-1)$.

Let $p$ be a prime number, $\mathbb Q_p$ be the completion of $\mathbb Q$ w.r.t $\,p$, denote $\xi_n$ primitive $n$-th root of unity in a fixed algebraic closure of $\mathbb Q_p$ with $p\nmid n$, we ...
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Absolute value of tensor product of fields

Suppose that we have the Laurent series fields $F_1:=\mathbb F_p((X))$ and $F_2=\mathbb F_p((Y))$. Equip $F_1$ with the $X$-adic multiplicative absolute value $|\cdot|_1$, i.e. define $|X|_1=\dfrac{...
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Under What assumptions on $p$, $\mathcal{O}_K^* \simeq \mathbb{Z}_p^{*} \oplus \mathbb{Z}_p^{*}$

Let $p$ be a fixed prime number and $\mathbb{Q}_p$ be the field of $p$-adic numbers and $K$ be an extension of degree $2$ of $\mathbb{Q}_p$. Let $\mathcal{O}_K$ be the ring of integers of $K$ and $\...
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Local Kronecker–Weber theorem implies the global one?

I was reading local Kronecker–Weber theorem implies global one in a course manual, but there's some parts I don't understand: Assume local Kronecker–Weber theorem, that is , every finite abelian ...
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If $\alpha^3+4k\alpha-k=0$ with $k\geq 1$ is odd, then $\mathbb{Q}(\alpha)$ has even class number.

This is an exercise found in course notes I have for an introductory course in class field theory and it has been bugging me a for long time. So $K=\mathbb{Q}(\alpha)$, with $\alpha$ a root of $f=X^3+...
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Extensions of Hilbert class fields

Let $K/F$ be a Galois extension of number fields. Is it true that the Hilbert class field $H_K$ of $K$ is an extension of the Hilbert class field $H_F$ of $F$ ? If the class number of $F$ is $h_F >...
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Why is $EK_w$ complete?

Let $K$ be a number field, $E$ a finite extension, $A$ the integral closure of $\Bbb Z$ in $K$, $B$ the integral closure of $A$ in $E$, let $w$ be an absolute value corresponding to a prime of $B$, ...
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1answer
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About the definition of local fields

I read in a book that the definition of local fields is: A field $K$ for which with respect to a discrete valuation $v$, the residue field is finite and $K$ is complete with respect to $v$. However, ...
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1answer
113 views

Artin symbol property

In Serge Lang's Algebraic Number Theory, Chapter 10, it defines Artin symbol as follows: Let $k$ a number field, $K/k$ be an abelian Galois extension, and $p$ be a prime of $k$ which is unramified ...
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$p >2$ is a prime, any facts about congruence relation between the class number of $Q(\sqrt p)$ and $Q(\sqrt{-p})$?

Let $p$ be an odd prime. This is a question about the class number of $Q(\sqrt p)$ and $Q(\sqrt{-p})$,which we denote by $h(p)$ and $h(-p)$ respectively. While doing my research on number theory I ...
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Vanishing of second Galois cohomology group

This most likely follows from a standard result but a lack of knowledge prevents me from seeing this. Let $K$ be a non Archimedean local field. Let $\Gamma$ be $\mathrm{Gal}(\bar{k}/k)$. Let $T$ be a ...
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Minkowski bound of $229$-th cyclotomic field

If $K = \mathbb{Q}\zeta_{229}$, the $229$-th cyclotomic field, Wwat is a good approximation for the bound $n$ such that every class in the ideal class group of $K$ contains an integral ideal of norm $...
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Non-Galois finite extensions of $F^{\operatorname{ur}}$

Let $F$ be a $p$-adic field with algebraic closure $\overline{F}$, and let $F^{\operatorname{ur}}$ be the maximal unramified extension of $F$. Let $E$ be a finite extension of $F$. I'm a bit rusty ...
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Is $h(n) < 2^n$ for all $n$? ($n$th cyclotomic field class number growth)

Is it ever the case that (for prime $n$) the $n$-th cyclotomic field class number, $h(n)$, is greater than $2^n$? (List of class numbers of $n$-th cyclotomic field for prime $n$). For instance, the $...