Questions tagged [class-field-theory]
The class-field-theory tag has no usage guidance.
382 questions
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Analogue of a ring extension splitting in the Kummer case
Background (the Kummer extension case)
Let $R$ be a complete regular local ring (it follws that it's a UFD) with a prime integer $p$ contained in the maximal ideal of $R$ (I'm mostly interested in $R=...
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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 ...
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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 ...
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Relation between the Hilbert Class polynomial of $\mathcal{O}_K$ and an order.
Hi all,
I have been looking at complex multiplication of elliptic curves for a course project in cryptography and the following question came up: Let $\mathcal{O}_K$ be the maximal order in $K$ ($K$ ...
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2
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Properties shared by number fields with the same normal closure?
While studying some class field theory there was a lot of talk on galois extensions. Of course. When talking about non-galois number fields, usually the text will quickly take the galois closure. At ...
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Class Field Theory for Imaginary Quadratic Fields
Let $K$ be a quadratic imaginary field, and $E$ an elliptic curve whose endomorphism ring is isomorphic to the full ring of integers of $K$. Let $j$ be its $j$-invariant, and $c$ an integral ideal of $...
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Algorithm for the class field tower problem?
This is a spur of the moment algebraic number theory question prompted by a side remark I made in a course I'm teaching:
Let $K$ be a number field. The (Hilbert) class field tower of $K$ is the ...
- 65.4k
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Extending methods from Lubin-Tate theory
The first lemma in Lubin-Tate theory says the following:
Let $K$ be a local field, $A$ its ring
of integers, and $f\in A[[T]]$ be such
that $f(0) = 0$, $f'(0)$ is a
uniformizer, and $f$ ...
- 243
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3
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Is there a notion of Galois extension for Z / p^2?
The above title is in fact a special case of what I want to ask.
Certainly we have a well defined notion of Galois extension for $ \mathbb{Q}_p $. The intersections of these extensions to the ring ...
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Is there an analog of class field theory over an arbitrary infinite field of algebraic numbers?
Recently, I found a paper by Schilling http://www.jstor.org/pss/2371426, which mentions that for certain infinite field of algebraic numbers there is an analog of class field theory. By infinite field ...
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ray class field of rational function field
Let $f \in \mathbf{F}_q[T]$ be irreducible. I know that the ray class field for $\mathrm{Cl}((f)) \cong (\mathbf{F}_q[T]/(f))^\times$ can be constructed by adjoining torsion points of a Carlitz module....
user19475
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2
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Elementary Aspects of Galois Deformation
Galois deformations are an important tool in Wiles' arsenal
for proving FLT. Are there any more elementary aspects (I'm
thinking of 1-dimensional Galois representations attached to
number fields) ...
- 32.6k
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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 ...
- 1,417
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Where can I find online copies of class field theory publications by Kronecker, Weber, Chevalley, Hasse, Hilbert, Takagi, etc?
I am writing an undergraduate thesis on local and global class field theory from a classical (i.e., non-cohomological) approach and am hoping to obtain copies of the early groundbreaking publications ...
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What's the Hilbert class field of an elliptic curve?
My question points in a direction similar to Qiaochu's, but it's not the same (or so I think). Let me provide you with a little bit of background first.
Let E be an elliptic curve defined over some ...
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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.
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Central simple algebras approach to class field theory, merits of
As noted earlier, I found reading Weil's book "Basic Number Theory" to be a harrowing experience, and I find his writing to be intrinsically hard to understand, though it is perfectly ...
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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 ...
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remark in milne's class field theory notes
In the introduction of his class field theory notes Milne mentions that some famous mathematicians failed to ask if the Artin isomorphism is canonical (between $Gal(L/K)$ and $C_m/H$ where $H$ is ...
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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 ...
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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?
user19475
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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 ...
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10
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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 ...
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5
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A problem of Shimura and its relation to class field theory
In Chapter II.10 of The Map of My Life, Goro Shimura mentions a certain problem:
The second topic concerns a polynomial $F(x)$ with integer coefficients. Take
$$
F(x) = x^3 + x^2 - 2x - 1,
$$
...
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5
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Tips on cohomology for number theory
I am curious about what is a good approach to the machinery of cohomology, especially in number-theoretic settings, but also in algebraic-geometric settings.
Do people just remember all the rules and ...
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6
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Reference for learning global class field theory using the original analytic proofs?
I'm wondering if anyone knows of a reference for learning global class field theory using the original analytic proofs developed in the 1920s and 1930s. Almost every book I can find either does local ...
- 15.4k
37
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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 ...
- 2,992
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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 ...
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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 $R=262537412640768744 - \epsilon$ ...
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Solvable class field theory
Is/should there be a theory of finite solvable extensions over a given base field? Could it be based on/use class field theory? Assume the base field isn't a local field.
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Neukirch's class field axiom and cohomology of units for unramified extension
This question may be too detailed but perhaps somebody knows the answer: Neukirch proofs in his algebraic number theory book in Chapter IV, Proposition 6.2, that his class field axiom implies that the ...
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What is the Hilbert class field of a cyclotomic field?
In the answers to Qiaochu's post on defining representations of finite groups over the algebraic integers, it came out that which fields a representation of a finite group is defined over might depend ...
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