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

learn more… | top users | synonyms

0
votes
1answer
30 views

Artin reciprocity theorem for Hilbert class field

In Cox's book "Primes of the form $x^2 + ny^2 $..." gives the following statement of Artin reciprocity theorem, for the Hilbert class field (i.e. maximal unramified Abelian extension) Artin's ...
1
vote
1answer
17 views

Why is $Gal(\Omega/k)$ a topological group under the Krull topology?

For an infinite Galois extension $\Omega/k$ the Krull topology on $G:=Gal(\Omega/k)$ is defined by taking as a basis for the neighbourhood of an element $\sigma \in G$ all cosets of the form $\sigma ...
2
votes
1answer
28 views

How does $ \text{Gal}(K / k) $ act on ideles?

Let $K/k$ be cyclic of degree $N$, Galois group $G$. I want to define some action of $G$ on the group of ideles $J_K$ which commutes with multiplication. A natural way to do this is to take each ...
2
votes
1answer
78 views

How does class field theory help us deduce the splitting of nonprincipal prime ideals?

I had a general question about the significance of global class field theory. One of the goals, as I understand, is to answer the following question: Given $L/K$ abelian, $g$ a divisor of $[L : ...
2
votes
0answers
17 views

norm map and local class field theory

Let $K$ be a local field, say a finite extension of $\mathbb{Q}_p$ (which is the purpose of my interest). Let $L$ be an unramified extension of $K$. Local class field theory asserts that there ...
1
vote
1answer
36 views

prime ideal of $\mathcal O_K$ splits completely in tower of Galois extensions

The following exercise is taken from D.A.Cox's book "Primes of the form $x^2 +ny^2...$" He uses this in order to prove some intermediate steps before the proof of the main theorem in chapter 5, which ...
0
votes
1answer
27 views

definition of Artin map

Let $L/K$ be an unramified Abelian extension. Then the Artin symbol $ \left ( (L/K) / \mathfrak p \right) $ is defined for all prime ideals $\mathfrak p$ of $\mathcal O_K$. (because in an Abelian ...
-1
votes
1answer
65 views

What does it mean for a prime ideal to divide a natural number m?

In Cassels and Frohlich (Algebraic Number Theory) Exercise 1, one is asked to derive some properties of the power residue symbol. It begins by stating the following: Let $m$ be a fixed natural ...
1
vote
1answer
27 views

Finite dimensional central division algebras over a finite extension of $\mathbb{F}_q(T)$

Over number fields, finite dimensional central division algebras are always cyclic algebras. So the construction of cyclic algebras is a nice recipe to create algebras, which exhausts all finite ...
1
vote
1answer
21 views

Existence of Conductor for Cyclotomic Extension (pg 200, Serge Lang A.N.T.)

Let $\zeta$ be a primitive $m$th root of unity, $m \not\equiv 2 \pmod 4$, and $K = \mathbb{Q}(\zeta)$. For $p$ prime and unramified i.e. $(m,p) = 1$, I know that the Artin symbol $(p, K/\mathbb{Q})$ ...
3
votes
1answer
65 views

The maximal unramified extension of a local field may not be complete

While reading my notes of a course in local class field theory, I arrived to a remark where it is said that given a complete discrete valuation field $K$, its maximal unramified extension $$K^{ur}= ...
1
vote
0answers
51 views

Unramified cubic extension of imaginary quadratic fields

Let $ K=\mathbb Q(\sqrt{-m})$ be an imaginary quadratic field with class number $ 6.$ Then by Hilbert class field theory and Galois correspondence it is known that $ K$ has a unramified cubic ...
0
votes
0answers
25 views

definition of the L-function $L(f, \chi, s): \mathbb{A}_K \rightarrow \mathbb{C}$, what is smoothness and what is $f$?

To summarize the question I'm going to ask: for those who have studied L-functions and class field theory, I am confused about the definitions of some things and haven't found a good reference for ...
1
vote
1answer
39 views

Discriminant of a Polynomial over a Local Field

I am trying to prove the local Kronecker-Weber theorem for tamely ramified abelian extensions $L|\mathbb{Q}_p$. At some point in the proof I need to show that $\mathbb{Q}_p(u^{1/e})$ is unramified ...
2
votes
1answer
40 views

Defining the Artin Map on the Ideles

Let $L/K$ be abelian. There is a natural way to define the Artin reciprocity map on the ideles using the notion of an admissible cycle. I don't want to go into the details of what that is right now, ...
1
vote
0answers
58 views

Algebraic proof of 2nd inequality of Global class field.

s there any 'explicit' proof, preferably without the use of cohomology, of second inequality of global field? By second inequality I mean $[C_K:N_{L/K}C_L]\leq [L:K]$ where $C_K$ is idele class group ...
2
votes
2answers
59 views

Where have I gone wrong in calculating norm residue symbol.

It is well known that the reciprocity map in local class field theory gives reciprocity map in Global class field theory. Namely, if $(\cdot,L_\beta/K_P):K_P^\times \rightarrow G(L_\beta/K_P)$ is the ...
3
votes
1answer
34 views

Are the discriminant of abelian cubic extensions of $\Bbb Q$ equal to the square of their conductor?

Here the conductor $N$ of an abelian extension $\Bbb Q \subset K$ is the smallest positive integer $N$ such that $K \subset \Bbb Q(\zeta_N)$. Thanks to class field theory there is an equivalence ...
1
vote
0answers
25 views

Does Idele group of norm 1 preserved by the norm?

I should explain my question in detail as of now I'm sure it makes no sense. Let $K$ be a global field (in particular I care about the characteristic $p$ case.) Then its Idele group $I_K$ has a ...
2
votes
1answer
62 views

Proof for a theorem on Cohomology by Tate

I am searching for a reference for the proof of the following theorem. Let $G$ be a finite group, let $C$ be a $G$-module, and let $u$ be an element of $\hat{H}^2(G,C)$. Assume that $\hat{H}^1(H,C) = ...
1
vote
0answers
50 views

Definition of module

In the book's, The theory of numbers, S. Iyanaga. Chater I, Cohomology of groups. What is the meaning of "module A"? Thank you all.
3
votes
1answer
117 views

Why do we have to work to prove the surjectivity of the local Artin map (Serge Lang A.N.T., Chapter XI Theorem 3)

I must be misunderstanding something about Artin reciprocity. Let $K/k$ be an abelian extension of number fields with Galois group $G$, $I_k$ the ideles of $k$, and $P$ a prime of $k$ (with $v$ a ...
17
votes
2answers
312 views

An interesting table of Prime Generating polynomials similar to $n^2+n+41$?

Here is some data on quadratic prime generating polynomials of a particular form. Kindly look at the questions given below it. $$\begin{array}{cccc} \text{#} & P(n)=an^2+bn+c\,; & d = ...
1
vote
0answers
69 views

Exercise 2.8 Cassels and Frohlich

I don't understand the discussion in exercise 2.8 of Cassels and Frohlich (page 352) beginning with "more generally". Why should it matter whether the formula for $c$ has a power of $-1$ in it if this ...
3
votes
0answers
35 views

Class number 1 for negative integers

If n is a positive integer then, $h(-4n) = 1 \iff n = 1,2,3,4,7$. The only proof I have seen of this is long and case wise. Is there any conceptual a priori reason to believe that these numbers are ...
1
vote
1answer
42 views

Is the group $I_K/K^{\ast}$ compact?

I have two question on adeles and ideles: $1)$Let $K$ be a number field. Is the group $I_K/K^{\ast}$ compact? Here $I_K$ is the idele group of $K$. $2)$ Also it will be helpful if someone explains ...
1
vote
0answers
36 views

Connected component of the Idele group

Let $K$ ba a number field with $r_1$ real embeddings and $r_2$ pairs of complex embeddings. Let $I_K$ be the group of ideles of $K$ and let $H$ be the connected component of identity. How to show that ...
3
votes
0answers
61 views

Reference request for Fourier analysis on local fields

I am studing Class field theory. I need a good reference books, notes e.t.c which explains the following topics : Ideles and ideals, haar volume measure and integration on local fields, Fourier ...
0
votes
0answers
43 views

Pure Cubic Fields

Let $K=\mathbb Q(\sqrt[3]m) $, $m $ cube free is a pure cubic field. Is there any relation between number of unramified quadratic extensions of $K$ and $2$ rank of the this field? Any help in this ...
6
votes
0answers
156 views

Generalized class group of $\mathbb Q(\sqrt{-5})$

I follow the notation of Georges Gras: Class Field Theory, some of which I recall for convenience; feel free to skip the following lines if you are familiar with the notation. Let $K$ be a number ...
10
votes
1answer
219 views

Cube roots of five

This is not really homework. I might be able to do this myself in time, from the methods in Ireland and Rosen. Note that every number has exactly one cube root $\pmod q$ for any prime $q \equiv 2 ...
5
votes
1answer
87 views

Unramification of Ideals in Pure Cubic fields

I need some explanation for this .Let $K=\mathbb Q{\sqrt[3]{m}} $ be a pure cubic field with non square element $\alpha $ in $K$ such that ideal $(\alpha) $ is an ideal square in K. Let $ ...
3
votes
1answer
90 views

Hilbert class field of cubic field

Let $K=\mathbb Q(\sqrt[3]7) $ be a pure cubic field with class number 3. I want know how to compute its Hilbert Class Field. I know that its degree of extension is 3. Thank You in advance.
2
votes
0answers
84 views

Show that i is an element of the p-adic integers if and only if p congruent to 1 mod 4

This exercise was given in a graduate course on Local Class Field Theory. We want to prove that $i\in \mathbb{Z}_p$ (the $p-$adic integers) if and only if $p\equiv 1 \mod 4$. For $\Rightarrow$, we ...
2
votes
1answer
62 views

Computing $\text{Gal}(\mathbb{Q}^{ab}/\mathbb{Q})$

Algebraic class field theory tells us that $\text{Gal}(\mathbb{Q}^{ab}/\mathbb{Q})$ is isomorphic to the group of connected components of the quotient $\mathbb{Q}^{\times}\backslash ...
3
votes
2answers
136 views

explicit example of computing ray class field for imaginary quadratic?

Given an imaginary quadratic number field K, we can get its ray class field mod some ideal $\mathcal{m}$ by adjoining the j-invariant of an elliptic curve with complex multiplication given by ...
7
votes
1answer
133 views

Characterizing a sequence of primes

This is an attempt to finish up Characterizing the primes which don't divide any Pell-Lucas number(s) For primes $p \equiv 3 \pmod 4,$ there is always some solution to $x^2 - 2 y^2 = \pm 1$ with ...
2
votes
0answers
45 views

Modular Forms Over Quadratic Number Fields

I'm trying to do a bit of reading and looking into mislay forms of weight one over quadratic number fields, but am finding it difficult to locate any books or papers. I've got a little material from ...
6
votes
2answers
148 views

history and/or motivation for cohomology in class field theory

I am currently learning (local) class field theory via group cohomology with Milne's notes. I have a number of questions about using group cohomology to prove the main statements of class field ...
3
votes
1answer
68 views

How to evaluate the quotient of Dedekind eta function in Pari/Gp

This expression I found in some research paper, which connects quotient of Dedekind eta function and ray class field of conductor N, which in turn gives the value of j-invariants. For $K=\mathbb ...
1
vote
0answers
11 views

Evaluating the quotient of Dedekind Eta function [duplicate]

This expression I found in some research paper, which connects quotient of Dedekind eta function and ray class field of conductor N, which in turn gives the value of j-invariants. For $K=\mathbb ...
2
votes
1answer
42 views

Evaluating $j$-invariant in PARI/GP.

Is there any command to evaluate $j$-invariant in PARI/GP? In Pari/Gp reference card there is $\operatorname{ellj}(x)$ function; but I am not understanding how to evaluate $j(i)$ or ...
4
votes
1answer
142 views

On Hilbert Class Polynomial

Is there any open source software which computes Hilbert Class polynomial of an imaginary quadratic fields? Thank you in advance
2
votes
2answers
91 views

Conductor of a ray class field.

I am not getting the definition of Conductor of a ray class field. I know the following definition Let $K$ be a number field. The theorems of class field theory tell us that given any modulus ...
4
votes
2answers
75 views

density theorems and class field theory

am I correct in thinking that the frobenius density theorem (it says that the Dirichlet density of the set of primes of K that split completely in an extension L is 1/[L:k]) is sort of one of the main ...
5
votes
1answer
56 views

Abelian extensions under inclusion, and their conductors

Suppose $K$ is a number field, and let $L$ and $L'$ be two abelian extensions of $K$, with conductors $C(L/K)=\mathcal{C}$ and $C(L'/K)=\mathcal{C}'$, respectively. Question: Is it true that the ...
0
votes
1answer
25 views

Question on the idele group and the topology on it

On page 68 of Nancy Childress' book "Class Field Theory", it says "We want to put a topology on $J_F$ that will make it a locally compact topological group." ($J_F$ is the group of ideles of $F$), ...
3
votes
0answers
52 views

Mapping of inertia group in local class field theory

Let $p$ be a rational prime, $K$ be a local field, $K(p) \mid K$ be the maximal $p$-extension of $K$ inside a given separable closure. Now let $I(K(p) \mid K)$ be the inertia group of $K(p) \mid K$ ...
1
vote
1answer
39 views

If $\zeta$ is a function of characters what does it mean for it to be regular?

This is from lemma 2.4.1 of Tate's thesis. Lemma 2.4.1: A $\zeta$-function is regular in the "domain" of all quasi-characters of exponent greater than $0$. proof: We must show that for each ...
9
votes
1answer
215 views

$\mathbb{Q}(i)$ has no unramified extensions

It is a classical result that every extension of $\mathbb{Q}$ is ramified. Put differently: there are no unramified extensions of $\mathbb{Q}$. The classical proof follows from the following two ...