Questions tagged [class-field-theory]

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57
votes
9answers
13k views

Learning Class Field Theory: Local or Global First?

I've noticed that there seem to be two approaches to learning class field theory. The first is to first learn about local fields and local class field theory, and then prove the basic theorems about ...
40
votes
2answers
6k views

Why is Class Field Theory the same as Langlands for GL_1?

I've heard many people say that class field theory is the same as the Langlands conjectures for GL_1 (and more specifically, that local Langlands for GL_1 is the same as local class field theory). ...
68
votes
10answers
15k views

Intuition for Group Cohomology

I'm beginning to learn cohomology for cyclic groups in preparation for use in the proofs of global class field theory (using ideal-theoretic arguments). I've seen the proof of the long exact sequence ...
64
votes
17answers
10k views

Shortest/Most elegant proof for $L(1,\chi)\neq 0$

Let $\chi$ be a Dirichlet character and $L(1,\chi)$ the associated L-function evaluated at $s=1$. What would be the 'shortest' proof of the non-vanishing of $L(1,\chi)$? Background: The non-vanishing ...
61
votes
8answers
11k views

Why are powers of $\exp(\pi\sqrt{163})$ almost integers?

I've been prodded to ask a question expanding this one on Ramanujan's constant $R=\exp(\pi\sqrt{163})$. Recall that $R$ is very close to an integer; specifically $R=262537412640768744 - \epsilon$ ...
25
votes
4answers
3k views

Why do congruence conditions not suffice to determine which primes split in non-abelian extensions?

How does one prove that the splitting of primes in a non-abelian extension of number fields is not determined by congruence conditions?
25
votes
9answers
12k views

Suggestions for good books on class field theory

Recently I tried to learn class field theory, but I find it is difficult. I have read the book "Algebraic Number Theory" by J. W. S. Cassels and A. Frohlich. In the book, the approach to class field ...
32
votes
1answer
4k views

Integers not represented by $ 2 x^2 + x y + 3 y^2 + z^3 - z $

EDIT, 9 March 2014: when I asked this in 2010, I did not have the courage of my convictions, and so did not ask for an if and only if proof, as Kevin Buzzard quite properly pointed out. Such problems ...
32
votes
4answers
2k views

$A_5$-extension of number fields unramified everywhere

So I was having tea with a colleague immensely more talented than myself and we were discussing his teaching algebraic number theory. He told me that he had given a few examples of abelian and ...
9
votes
2answers
764 views

transfer kernels and the Schur multiplier

Let $\Gamma$ be a finite $2$-group, and let $G$ be any subgroup of index $2$. Moreover, let Ver$: \Gamma/\Gamma' \to G/G'$ denote the group theoretical transfer, and let $M(\Gamma)$ be the Schur ...
4
votes
0answers
322 views

Euler Systems and Coleman’s Conjecture

I’m trying to work on Coleman’s conjecture for abelian extensions of imaginary quadratic fields. I’ve read most papers by Seo regarding circular distributions. However, I’m a still confused about what ...
70
votes
2answers
9k views

Please check my 6-line proof of Fermat's Last Theorem.

Kidding, kidding. But I do have a question about an $n$-line outline of a proof of the first case of FLT, with $n$ relatively small. Here's a result of Eichler (remark after Theorem 6.23 in ...
31
votes
5answers
7k views

A reference for geometric class field theory?

The classic reference of this topic is Serre's Algebraic Groups and Class Fields. However, many parts of this book use Weil's language, which I find quite hard to follow. Is there another reference ...
38
votes
3answers
4k views

Topological Langlands?

In a workshop about the geometry of $\mathbb{F}_1$ I attended recently, it came up a question related to a mysterious but "not-so-secret-anymore" seminar about... an hypothetical Topological Langlands ...
37
votes
1answer
2k views

Degree 17 number fields ramified only at 2

The number $17$ is the smallest odd number that occurs as the degree of a number field $K/\mathbb{Q}$ for which the only finite prime that ramifies is $2$. The non-existence for $n < 17$ follows ...
49
votes
13answers
6k views

Erratum for Cassels-Froehlich

Edit 25 April 2010: I have a physical copy of the new printing of the book. I can only assume the LMS is now selling it (but have no details). IMPORTANT EDIT: THE RESULTS ARE IN! Ok, the deadline has ...
13
votes
5answers
2k views

What is the "reason" for modularity results?

The question is a little wishy-washy, but I take my cues from other popular questions that relate to the philosophy behind the mathematics as Why do Groups and Abelian Groups feel so different? . I ...
19
votes
3answers
2k views

Where does the principal ideal theorem (from CFT) go?

My impression is that one of the celebrated results of class field theory the principal ideal theorem namely that given a number field $K$ and its maximum unramified abelian extension L, every ideal ...
20
votes
1answer
1k views

Class number parity in pure cubic number fields

Consider the family of pure cubic number fields $K = {\mathbb Q}(\sqrt[3]{m})$ for $m = a^3 \pm 3$. Proposition. If $4 \mid a$ and $m$ is cubefree, then the class number of $K$ is even. Proof. Let $...
11
votes
3answers
2k views

What makes Geometric CFT easier than CFT?

I've been reading: math.stanford.edu/~conrad/249BPage/handouts/geomcft.pdf in an attempt to shed some geometric light on class field theory. The last paragraph there reads: In case the ground field $...
14
votes
5answers
3k views

Given a number field $K$, when is its Hilbert class field an abelian extension of $\mathbb{Q}$?

Given a number field $K$, when is its Hilbert class field an abelian extension of $\mathbb{Q}$? I am going to be on the road soon, so pleas don't be offended if I don't respond quickly to a comment.
20
votes
1answer
1k views

Can one prove complex multiplication without assuming CFT?

The Kronecker-Weber Theorem, stating that any abelian extension of $\mathbb Q$ is contained in a cyclotomic extension, is a fairly easy consequence of Artin reciprocity in class field theory (one just ...
17
votes
3answers
1k views

The Teichmüller's algebraic interpretation of $H^3$ in group cohomology

In the book "Cohomology of Groups" of Kenneth S. Brown, it is told in the introduction that Teichmüller arrived to $H^3$ in an algebraic context, i.e. that Teichmüller worked with an ...
14
votes
2answers
990 views

Class groups in dihedral extensions - some sort of Spiegelungssatz?

Let $p$ be an odd prime and let $F/\mathbb{Q}$ be a Galois extension with Galois group $D_{2p}$, let $K$ be the intermediate quadratic extension of $\mathbb{Q}$, and $L$ an intermediate degree $p$ ...
23
votes
1answer
1k views

Any open Langlands Conjectures for GL_1?

Are there any general conjectures/properties (in the Langlands Program) for automorphic representations of $GL_n$ which are still open for $n=1$?
8
votes
1answer
876 views

Numbers integrally represented by a ternary cubic form

Given integers $a,b,c,$ and cubic form $$ f(a,b,c) = a^3 + b^3 + c^3 + a^2 b - a b^2 + 3 a^2 c - a c^2 + b^2 c - b c^2 - 4 a b c $$ $$ f(a,b,c) = \det \left( \begin{array}{ccc} a & b &...
9
votes
4answers
2k views

Origins of functional field arithmetic

Background: By function field, we mean a finite extension of the field of rational functions of one variable over a finite field with $p$ elements. Classfield theory for function fields was ...
8
votes
3answers
2k views

Maximal (non-abelian) extensions of number fields unramified everywhere

Hello! Let $K$ be a number field. All abelian unramified extensions are contained in the Hilbert class field which is a finite extension 'maximal' with respect to this property. For general ...
6
votes
1answer
943 views

Set of quadratic forms that represents all primes

A SPECIFIC CASE: Any prime number can be classified as either $p \equiv 1 \pmod 3$ or $p \equiv 2 \pmod 3$. If $p = 3$ or $p = 1 \pmod 3$, then the prime $p$ can be represented by the quadratic form $ ...
1
vote
2answers
1k views

2-class group of a quadratic imaginary extension

Let $p\equiv 5 [8]$ be a prime number, and consider $K=\mathbb{Q}(\sqrt{-p})$. I would like to check that the $2$-Sylow subgroup of the class group $C_K$ has order $2$ (I'm pretty sure it's true). ...
9
votes
1answer
448 views

Globalizing local field extensions with controlled ramification

Let $K_1/k_1, \ldots, K_r/k_r$ be "separable cyclic" extensions of degree $n$ where each $k_i$ is a local field of characteristic 0 (archimedean or not). By separable cyclic of degree $n$, I mean the ...
12
votes
0answers
742 views

What numbers are integrally represented by $4 x^2 + 2 x y + 7 y^2 - z^3$

This is related to my first MO question and Kevin Buzzard's conjecture at Integers not represented by $ 2 x^2 + x y + 3 y^2 + z^3 - z $ In December 2010 my question appeared in the M.A.A. Monthly, ...
7
votes
1answer
368 views

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 $\mathbb{Q}(\zeta_{24})/\mathbb{Q}(\sqrt{-6})$ ...
5
votes
1answer
576 views

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 ...
4
votes
1answer
175 views

A Kummer exact sequence involving $\mu_\infty$

Let $k$ be a number field. We have the well-known Kummer exact sequence of etale sheaves on $\mathrm{Spec}\, k$: $$1 \rightarrow \mu_n \rightarrow \mathbb{G}_m \rightarrow \mathbb{G}_m \rightarrow 1.$$...
8
votes
1answer
504 views

how to compute Hilbert class field of $\Bbb Q(\zeta_{23})$?

I want to construct the Hilbert class field of $K=\Bbb Q(\zeta_{23}).$ I have no clue how to construct it except that I know that $[H(K):K]=3$ from Sage. Any references or comments are appreciated.
6
votes
1answer
671 views

Parity of class number of pure cubic fields

A pure cubic field is an algebraic number field of the form $K = \mathbb{Q}(\theta)$ with $\theta^3 = m$, $m \neq \pm 1$. What can be said about the parity (odd or even) of the class number of a pure ...
6
votes
1answer
291 views

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 $[...
5
votes
1answer
232 views

Positive Primes represented by an indefinite binary form, reducing poly degree from 8 to 4

In his lovely answer at Positive primes represented by indefinite binary quadratic form Noam found that a (positive) odd prime $p$ is represented by the indefinite form $x^2 + 13 x y - 9 y^2$ if and ...
4
votes
1answer
296 views

Examples of norm forms where the numbers represented can be readily described

In case of impatience: the question here is a request for examples, especially degree six or seven where the norm form might represent some prime$p,$ then some $q^2$ but not $q,$ then some $r^3$ but ...
4
votes
0answers
222 views

Polynomial equations in many variables have solutions (Lang 1952 paper)

I am trying to understand the proof of the following result: Suppose $F$ is a function field in $k$ variables over an algebraically closed field. Let $f_1,...,f_r \in F[x_1,...,x_n]$ be ...
2
votes
0answers
780 views

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 $K$,...
2
votes
0answers
115 views

Does Langlands use the geometric Frobenius or the classical Frobenius in his papers?

In several of Langlands' papers: Representations of Abelian Algebraic Groups, On Artin's L-functions, On the Functional Equation of Artin's L-functions, Langlands takes a finite Galois extension $K/F$ ...
4
votes
1answer
606 views

Why is Kronecker's Jugendtraum only for abelian extensions?

Why is Kronecker's Jugendtraum only for abelian and not for more general extensions of number fields? Wikipedia, Hilbert's Twelfth Problem.
4
votes
1answer
114 views

Norm groups of number fields

I came across this proposition in an article about genus class fields. I have a few questions about the parts that I have underlined in red. I don't understand why the norm map $N_{H/K}: I_H \to P_K$ ...
4
votes
0answers
307 views

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 ...
0
votes
0answers
75 views

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 ...