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2
votes
0answers
159 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 ...
2
votes
1answer
83 views

A certain idele class character

Let $E/K$ be a cubic extension of number fields, $\nu$ be a Grossencharacter of the idele class group $\mathbb{I}_{E}/E^{\ast}$ such that $\nu^2$ is trivial and $\nu$ restricted to the idele class ...
14
votes
0answers
487 views

Positive binary quadratic form plus univariate monic cubic (giving Hilbert class field)

We have the Lucas numbers, $$ L_1 = 1, \; L_2 = 3, \; L_3 =4, \; L_4 = 7, L_5 = 11, \; L_{n+2} = L_{n+1}+ L_n \; . $$ Question: is it the case that $$ f(x,y,z) = 4 x^2 + 3 x y + 9 y^2 + z^3 + 3 z ...
1
vote
0answers
95 views

Skew symmetry for the Hilbert symbol

Let $K$ be a local field containing the group $\mu_n$ of $n$th roots of 1 and the $\theta_K:K^*\to G_K^{ab}$ be the reciprocity map. The we know that the Hilbert symbol $$K^*\times K^*\to \mu_n$$ ...
4
votes
3answers
343 views

Textbook request for class field theory [duplicate]

I am studying class field theory. I need good reference books, notes, or other materials which explain the following topics: ideles and ideals, Haar measure and integration on local fields, Fourier ...
0
votes
0answers
86 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
0answers
63 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
1answer
76 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
0answers
117 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
2answers
442 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
1answer
246 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
0answers
105 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
0answers
196 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
1answer
185 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
3answers
458 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
0answers
474 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
1answer
170 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
0answers
126 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
0answers
155 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
1answer
317 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
1answer
383 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
0answers
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
0answers
174 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
0answers
52 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
0answers
50 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
0answers
179 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
1answer
234 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
0answers
262 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
0answers
79 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
2answers
332 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
1answer
381 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
1answer
174 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
2answers
293 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
2answers
299 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
0answers
253 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
2answers
259 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
1answer
327 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
2answers
373 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
1answer
471 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
0answers
119 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
1answer
556 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
0answers
513 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
0answers
128 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
1answer
284 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
1answer
409 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
0answers
254 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
2answers
356 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
1answer
658 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
2answers
922 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 ...
10
votes
0answers
642 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, ...