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
0answers
14 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 ...
0
votes
0answers
14 views

Local Reciprocity Map

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 ...
15
votes
1answer
236 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
33 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
30 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
0answers
22 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 ...
0
votes
0answers
20 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
39 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 ...
1
vote
0answers
17 views

Pure Cubic Fields

Let $K=\mathbb Q(\sqrt[3]m) $, $m $ cube free is a pure cubic field. Then are we able to get number of unramified quadratic extensions of this field? Any help in this direction is useful. Thank you ...
6
votes
0answers
106 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 ...
8
votes
1answer
180 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 ...
4
votes
1answer
72 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
68 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.
1
vote
0answers
74 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
56 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
86 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
132 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
40 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 ...
5
votes
2answers
84 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
49 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
35 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 ...
3
votes
1answer
91 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
80 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
58 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
49 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
24 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
41 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
37 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
167 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 ...
3
votes
0answers
64 views

Genus field = Hilbert Class Field (Cox exercise 6.15)

Prove that the genus field of an imaginary quadratic field of $K$ equals its Hilbert Class Field if and only if for primitive forms of discriminant $d_k$, there is only one class per genus. ...
1
vote
1answer
27 views

ramification of valuations

By a “prime” of K (number field), we mean an equivalence class of nontrivial valuations on K. What does it mean for a finite prime p to ramify in an extension L of K? I'm reading these notes ...
1
vote
1answer
59 views

infinite primes

I am reading these notes by milne. http://www.jmilne.org/math/CourseNotes/CFT310.pdf. In it page 4 example 0.5 for instance he talks about "both infinite primes." Does anyone know what this means? ...
3
votes
2answers
95 views

Which remarkable properties does the Hilbert Class Field have?

Let $L$ be the Hilbert Class Field of $K$, then: $Gal(L/K) \cong Cl(K)$ by Artin reciprocity, where $Cl(K)$ is the class group of $K$. though being Galois is not transitive in general, we ...
6
votes
1answer
63 views

Question about a remark in Serre's Local Fields

I am reading Serre's Local Fields. In Section V.4, Serre considers a finite totally ramified extension of local fields $L/K$ with the residue field $\bar{L}=\bar{K}$ a perfect field. For $\bar{K}'$ a ...
3
votes
1answer
68 views

Interpretation of $S$-ideal class group

I have a problem understanding the interpretation of the ideal class group in the case of restricted ramifiction. Let $k$ be a number field and $S$ a set of primes of $k$. Then $k_S$ denotes the ...
7
votes
1answer
96 views

Volume of first cohomology of arithmetic complex

Let $K$ be a number field and consider the Arithmentic complex $\Gamma_{Ar}(1)^\bullet$ be defined by $$\begin{array} A\Bbb R^{r_1+r_2} & \stackrel{\Sigma}{\longrightarrow} & \Bbb R \\ ...
3
votes
1answer
112 views

Algorithm to find solutions $(p,x,y)$ for the equation $p=x^2 + ny^2$.

As the classical book of David Cox argues, Assume the conditions are satisfied and $p$ can be represented as $x^2 + ny^2$. What would be a way to find solutions $(p,x,y)$ efficiently? Ideally, one ...
1
vote
0answers
68 views

Tricks to find the Hilbert Class field of a quadratic extension?

Let $L$ be the Hilbert Class Field of $K=\mathbb{Q}(\sqrt{-d})$. I already know, via Artin reciprocity, that $Gal(L/K) \cong CL(K)$. Another theorem (Cox 9.30) says that: $Gal(L/\mathbb{Q}) \cong ...
3
votes
0answers
67 views

class field theory via schemes?

I know there is a close relationship between algebraic number theory and algebraic geometry. And in particular the theory of schemes is of many uses in algebraic number theory. Since I think the peak ...
3
votes
0answers
48 views

$P_{K,1}(\mathfrak m)\subset \operatorname {ker} \Phi_{\mathfrak m,L|K} \subset \operatorname {ker} \Phi_{\mathfrak m,M|K}$ imples $M \subset L$

Let $K$ be a number field and $L, M$ finite abelian extensions. Let $\mathfrak m$ be a modulus. Consider the two Artin maps $ \Phi_{\mathfrak m,L|K}$ and $ \Phi_{\mathfrak m,M|K}$. Let ...
2
votes
0answers
37 views

Confusing application of power residue reciprocity in Milne's CFT

Hey I am trying to figure out the details of the proof of Theorem 5.14 (p.246) in Milne's CFT (see here). I hope somebody is familiar with this. But let me sketch the proof and what I don't ...
0
votes
0answers
92 views

Field of finite transcendence degree over the field of rational numbers

Let $K$ an algebraically closed field with characteristic zero. Is it posible to find a subfield $F\subset K$ wich is a field of finite transcendence degree over the field $\mathbb{Q}$? Thanks a lot ...
1
vote
0answers
28 views

Intersections of all open subsets of finite index in the idele group

Let $K$ be a number field. Why is the intersections of all open subgroups of finite index in the idele group $\mathbb I_K$ equal to $\overline{K^{\times}(K^{\times}_{\infty})^0}$? Also, I'm having a ...
2
votes
1answer
97 views

Fields and proper subfields. [duplicate]

Specific question: Let $F$ be a field and assume that $\mathbb{Q}$ is a proper subfield of $F$. Can $F$ be isomorphic to $\mathbb{Q}$? Studying the foundaments of field theory I have to ask: Can ...
5
votes
2answers
121 views

Importance of continuity of Galois representations

So for a one dimensional Galois representation $\rho: G_{\Bbb Q} \to \mathbb C^{\times}$, I know that it must factor through the abelianization of $G_{\Bbb Q}$, which by the Kronecker-Weber theorem is ...
1
vote
1answer
54 views

Galois extension of an imaginary quadratic field

This is an exercise problem from the book "Primes of the form x^2+ny^2: Fermat, Class field Theory and Complex multiplication" Question: Let $K$ be an imaginary quadratic field, and let $K\subset L$ ...
2
votes
1answer
124 views

Hilbert class field of $\mathbb Q(\sqrt{-5})$ and $\mathbb Q(\sqrt{-10})$

This might be a very silly doubt, as I am making an attempt to understand Hilbert Class field, I would like to know this. In Cohen,"Advanced topics in computational number theory" Hilbert class field ...
9
votes
1answer
80 views

Intersection of class number one fields

Let $F$ and $K$ be two number fields with class number one. How can one prove that the class number of $F \cap K$ is also equal to one. I have been trying to prove something like the intersection of ...
7
votes
1answer
107 views

What do we know about the class group of cyclotomic fields over $\mathbb{Q}$?

Motivated by this question, I am curious how one can characterize primes that splits completely in the Hilbert class field of $\mathbb{Q}(\zeta_q)$, where $q$ is a prime. Then I realize how much I ...