The class-field-theory tag has no wiki summary.

**0**

votes

**0**answers

84 views

### Reciprocity laws in different dimensions

Let $M/L/Qp$ be a finite galois abelian extension of local fields and define
$\mathcal{M}=M\{\{T\}\}=\{\sum_{i\in \mathbb{Z}}a_iT^i:a_i\in M,\min_{i\in \mathbb{Z}}, v(a_i)>−\infty , \lim_{i\to ...

**1**

vote

**0**answers

58 views

### Relation between 1-dimensional and 2-dimensional reciprocity maps

Let $M/L/\mathbb{Q}_p$ be a finite galois abelian extension of local fields and define
$\mathcal{M}=M\{\{T\}\}=\{\sum_{i\in \mathcal{Z}}a_iT^i : a_i\in M, \min_{i\in \mathcal{Z}} v(a_i)>-\infty, ...

**1**

vote

**1**answer

69 views

### Equivalence of definitions of the Milnor $K$-groups

In Kurihara's paper: "The exponential homomorphisms for the Milnor $K$-groups and an explicit reciprocity law" he difines, in the first page, the $q$-th Milnor K-group for the ring $R$ as
...

**3**

votes

**0**answers

113 views

### Conics over number fields

I am looking for a reference for the following fact.
Let $k$ be a number field and let $S$ be a finite set of places of $k$ of even cardinality. Then there exists a unique conic $C$ over $k$ such ...

**7**

votes

**2**answers

431 views

### Galois groups and prescribed ramification

What is known about finite groups $G$ for which there exists a Galois extension $K$ of $\mathbb{Q}$ ramified only at $2$ such that $\text{Gal}(K/\mathbb{Q}) \cong G$ ? More generally, which groups can ...

**3**

votes

**1**answer

236 views

### Theorem 7b of Serre's “Propriétés galoisiennes des points d'ordre fini des courbes elliptiques”

Could someone please point me towards a proof of the statement in the second paragraph, in the proof of Theorem 7b of Serre's Propriétés galoisiennes...? The statement is as follows:
Let $F$ and $F'$ ...

**0**

votes

**0**answers

100 views

### Hilbert Class fields and Pure cubic fields

I want to know about Hilbert class field theory for pure cubic fields. Which is the best source for this? Right now I am reading book by D.A. Cox "Primes of the form $x^2+ny^2$" Fermat,Class field ...

**10**

votes

**0**answers

180 views

### Cubic fields correspond to $3$-torsion ideals in quadratic fields, or to order $3$ characters of quadratic class groups?

I was watching Dick Gross's laudation for Manjul Bhargava, followed up by one of Bhargava's talks, and I realized I was confused about something.
Bhargava says (around 21 minutes) that the orbits of ...

**3**

votes

**1**answer

177 views

### Cyclotomic character in class field theory

Let $K$ be an extension of $\mathbb{Q}_p$.
By local class field theory, the $p$-adic cyclotomic character $\mathrm{Gal}_K \rightarrow \mathbb{Z}_p^\times$ corresponds to a character $\chi : K^\times ...

**10**

votes

**3**answers

443 views

### Ideal classes fixed by the Galois group

Let $K$ be a number field and let $G$ be the group of automorphisms of $K$ over $\mathbf Q$. The group $G$ acts in a natural way on the ideal class group of $K$. I would like to know if there are any ...

**25**

votes

**0**answers

439 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 ...

**5**

votes

**1**answer

167 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 ...

**2**

votes

**0**answers

122 views

### Global Artin reciprocity law from Local class field theory

Let $K=\mathbb F_q((t)), p -$ prime ideal in $K$, $\psi_p$ be the local Artin map$K_p^* \to Gal(K_p^{ab}/K_p)=G_p \subset Gal(K^{ab}/K)$. Then I define global Artin map $\psi_K$as product of $\psi_p$, ...

**1**

vote

**0**answers

154 views

### Reference for Local class field theory via witt vectors

I would like to find some books or lecture notes on geometric local class field theory via Witt vectors. I can't find any good paper on this subject.All approaches in the books to local class field ...

**6**

votes

**1**answer

315 views

### Class groups of orders

In Cox's book "Primes of the form $x^2 + ny^2$", he proves that in a quadratic imaginary field $K$, if $\mathcal O$ is an order of conductor $f \in \mathbb Z$, we have that the class group ...

**7**

votes

**1**answer

377 views

### Hilbert Class Field Galois over Q?

So if we have a Galois extension $K/\mathbb{Q}$, then the Hilbert Class Field $H$ of $K$ is certainly Galois over $\mathbb{Q}$. But is the converse true? I know many examples of nongalois ...

**0**

votes

**0**answers

25 views

### necessity of divisibility critria for class number of a real quadratic field

Are all the conditions for divisibility of class number of a given real quadratic field by a given number sufficient only?
e.g. it is well known class number is divisible by 2 if the discriminant ...

**2**

votes

**0**answers

160 views

### Ring class fields of orders

The theory of ring class fields corresponding to orders in imaginary quadratic fields is introduced by, say, [1]. But I'm reading an article [2], in which ring class fields corresponding to orders in ...

**0**

votes

**0**answers

44 views

### sufficiency of the relative discriminant to be a square of an ideal for an unramified quadratic extension

Is it sufficient for a quadratic extension of a cubic number field to have a relative discriminant as a square of an ideal for being unramified extension (excluding primes dividing 2 for the sake of ...

**0**

votes

**0**answers

49 views

### divisor class group with modulus

Let $C$ be a smooth projective curve over a field $k$ and $S \subset C$ a finite number of points. A modulus is simply a divisor supported on $S$. What is the divisor class group with modulus?
I ...

**6**

votes

**0**answers

177 views

### n-dimensional local fields

Recently, I hear the concept of $n$-dimensional local fields.
It is defined inductively as follows.
(1) a $0$-dimensional local field is a finite field.
(2) an $n$-dimensional local field is a ...

**3**

votes

**1**answer

231 views

### Unramified extension and class field theory

I am not sure this question is proper for this site, but there is no other places that I can get an answer. So if anyone can give an answer for this, it would be very helpful to me.
Let $F$ be a ...

**5**

votes

**0**answers

255 views

### maximal abelian extension of quadratic extension of $\mathbb Q_p$

I read this article "Local class field theory via Lubin-Tate theory" http://arxiv.org/pdf/math/0606108v2.pdf. And I want to find the maximal abelian extensions for quadratic extensions of $\mathbb ...

**1**

vote

**0**answers

78 views

### Reference request: modern idelic formulation of class field theory for function fields

I am looking for a modern reference for idelic class field theory for function fields over finite fields including proofs.
For example, the book Algebraic Number Theory of Cassels-Frohlich does not ...

**4**

votes

**2**answers

328 views

### On class numbers $h(-d)$ and the diophantine equation $x^2+dy^2 = 2^{2+h(-d)}$

Given fundamental discriminant $d \equiv -1 \bmod 8$ such that the quadratic imaginary number field $\mathbb{Q}(\sqrt{-d})$ has odd class number $h(-d)$. Is it true that one can always solve the ...

**4**

votes

**1**answer

365 views

### About principal ideal theorem in number fields

I usually consider a cyclic extension $K$ of degree an odd prime $p$ over the rational field $\mathbf{Q}$.
In this case, there is a well-known result that "every ambiguous class in the class group ...

**0**

votes

**1**answer

163 views

### Idelic Artin map

I want to study the idelic Artin map for the, say, 7th cyclotomic field explicitly. In other words I want to see how a particular idele can be constructed which gets sent to the identity element in ...

**4**

votes

**2**answers

286 views

### How does Tate cohomology fit into a derived categories framework?

I've read through one class field theory text after another, but there's something very non-intuitive for me about cohomology that makes it hard for me to understand why Tate cohomology was invented.
...

**1**

vote

**2**answers

296 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).
...

**1**

vote

**0**answers

240 views

### Ray class field and ring class field

Let $K$ be quadratic number field and let $O$ be an order of $K$. A modulus $m$ of $K$ is a formal product $m_0\cdot m_\infty$ of finitely many finite primes $m_0$ and finitely many infinite real ...

**3**

votes

**2**answers

256 views

### Galois cohomology of the field of Laurent series

Let $k$ a separably closed field. Do we have that $k((t))$ is of cohomological dimension one?

**3**

votes

**1**answer

311 views

### Fields whose embeddings into the complex numbers are invariant under complex conjugation

Is there a general notion/description of fields $K$ such that the image of any embedding $K \hookrightarrow \mathbb{C}$ is invariant under complex conjugation, thus inducing an involution on $K$ which ...

**6**

votes

**2**answers

368 views

### the global m-th power reciprocity law and Quartic Reciprocity Law

I'm reading Cox "Primes of the form $x^2+ny^2$". And I read a chapter about the global m-th power reciprocity law. Now I'm not able to prove the quartic and cubic reciprocity laws. Where can i find ...

**7**

votes

**1**answer

464 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 ...

**1**

vote

**0**answers

117 views

### Finite extensions of residue fields of Henselian DVRs

Let $K$ be an Henselian discrete valuation field such that its completion is separable over $K$. Let $F$ be its infinite residue field. Is it true that a finite extension of $F$ is a simple extension ...

**7**

votes

**1**answer

549 views

### How did Takagi prove Kronecker's Jugendtraum for Q(i)?

In Noah Snyder's historical undergraduate thesis on Artin L-Functions, it mentions that Takagi proved Kronecker's Jugendtraum in the case of Q(i) in his doctoral thesis. Since I don't know how to get ...

**3**

votes

**0**answers

491 views

### Characterizing primes that split completely vs. primes with a given splitting behavior

Given a finite abelian extension of number fields $L/K$, the prime ideals $\mathfrak{p}$ in $O_K$ split into primes $\mathfrak{P}$ in $O_L$. The number of primes $\mathfrak{p}$ splits into is ...

**4**

votes

**0**answers

126 views

### Reciprocity Map and Cycle Class Map

This might be a very naive question but here it goes. Let X be a smooth variety of dimension d over a p-adic field. We have the n part of the rerciprocity map:
$rec/n: SK_1(X)/n \to \pi^{ab}_1(X)/n$
...

**3**

votes

**1**answer

278 views

### Abelianized fundamental group of a curve over a finite field

Let $X$ be a smooth, projective, and geometrically connected curve over a finite field $\mathbb{F}_q$ and fix a geometric point $\overline{x} : \text{Spec } \overline{\mathbb{F}_q} \to X$. Then there ...

**3**

votes

**1**answer

399 views

### Non-cyclotomic abelian extensions

Suppose $L|\mathbb{Q}$ is an abelian extension of number fields. Then, all the roots of unity are certainly contained in the maximal abelian extension $L^{ab}$ of $L$. Why is it obvious that if $L \ne ...

**4**

votes

**0**answers

250 views

### Abelian cubic extensions of Q[i],

Recently I was considering cubic extensions $K/Q$ that have discriminant negative of a perfect square. Classifying these curves reduces to solving a Diophatine equation of the form $4a^3+27b^2=c^2$ ...

**4**

votes

**2**answers

355 views

### Intersection of Hilbert class fields of imaginary quadratic fields

In this question Hilbert class field of Quadratic fields it is mentioned that if $d\equiv 1 \mod 4$ then the Hilbert class field of $\mathbb{Q}(\sqrt{-d})$ contains $\mathbb{Q}(i,\sqrt{d})$.
Could ...

**9**

votes

**1**answer

650 views

### The Class Number One Problem for Real Quadratic Fields

An approach to the Gauß class number one problem for imaginary quadratic fields is to determine the integral points on the modular curve $Y_{nonsplit}(n)$ for a suitable $n$. Here follows a quick ...

**12**

votes

**2**answers

907 views

### Elliptic Curves with CM and Class Field Theory

Let $K$ be an imaginary quadratic field with Hilbert class field $H$, and let $E$ be an elliptic curve defined over $H$ with complex multiplication by the ring of integers $O_K$ of $K$. It is known ...

**9**

votes

**0**answers

611 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, ...

**14**

votes

**5**answers

985 views

### What is the “ray” in ray class group?

I have never seen any algebraic number theory book discuss the origin of the term "ray class group." Does anyone know where the word "ray" comes from in this context? I always thought it might be a ...

**9**

votes

**0**answers

581 views

### How does one understand geometric CFT in terms of modularity?

I have recently asked a question in a similar vein:
What makes Geometric CFT easier than CFT?
but I'm afraid I wasn't quite ripe to ask it yet. I have since consulted with the following sources:
...

**6**

votes

**3**answers

873 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 ...

**20**

votes

**1**answer

926 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 ...

**3**

votes

**0**answers

216 views

### Formal non-CM in local fields

An elliptic curve $E$ with complex multiplication by an imaginary quadratic field $F$ has $\ell$-adic Galois representations that essentially encode the class field theory of $F$ - in other words, the ...