3
votes
2answers
48 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
116 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 ...
1
vote
0answers
29 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 ...
4
votes
2answers
48 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 ...
2
votes
1answer
31 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
64 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
4
votes
2answers
55 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
42 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 ...
3
votes
0answers
33 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 $K(p) \mid K$ ...
9
votes
1answer
146 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
52 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. ...
3
votes
2answers
85 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 ...
3
votes
1answer
59 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
93 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
101 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
57 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
49 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
46 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
35 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 ...
1
vote
0answers
21 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 ...
5
votes
2answers
103 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 ...
9
votes
1answer
71 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
101 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 ...
0
votes
0answers
30 views

$p > 2$ and ramification of archimedean places

Fix a rational prime $p$. I know that for a $p$-extension (ie. a Galois extension of degree a power of $p$) of an algebraic number field $k$, some places can not ramify: complex places cannot ...
2
votes
0answers
51 views

Weil group of Lubin-Tate extension

I'm reading this article of local class field theory http://arxiv.org/pdf/math/0606108v2.pdf. I can't understand the proof of prop 4.7(ii)(p. 8), more precisely last sentence. 'It is bijective because ...
3
votes
1answer
105 views

Fundamental theorem of local class field theory

Let $L/K$ be an abelian extension and $\frak p$ a place of $K,$ Let $\theta_\frak p$$ :K_\frak p^*\twoheadrightarrow$$ G(\frak p)$ the $\frak p$-local Artin map (defined in Algebraic Number Fields ...
7
votes
1answer
115 views

Infinite $p$-extension contains $\mathbb{Z}_p$-extension

Does the Galois group of every infinite $p$-extension $K$ of a number field $k$ contain a (closed) subgroup such that the quotient group is isomorphic to $\mathbb{Z}_p$? My feeling is "yes", but I'm ...
1
vote
0answers
103 views

Hilbert class field

I got this doubt after going through tables given by Henri Cohen: Advanced topics in Computational number theory on "Hilbert Class Field of Imaginary quadratic field". pg no.539-542, sec 12.1 and a ...
1
vote
0answers
170 views

Hilbert class field of $\mathbb Q(\sqrt{-14})$

The Hilbert Class Field of $K=\mathbb Q(\sqrt{-14})$ is $L=K(\alpha)$, where $\alpha=\sqrt{2\sqrt{2}-1}$. Then how does one obtain the Hilbert Class field of discriminant as $X^4-X^3+X+1$.
2
votes
1answer
33 views

Lemma for the construction of the reciprocity map

I do not understand the highlighted part in the following proof, namley that $N(\tilde x)=1$. To give some context, this proof is taken from Neukirch's Algebraic Number Theory, where $\tilde K$ ...
6
votes
1answer
267 views

Ring class field of $\mathbb{Q}(\sqrt{-19})$

I am looking an explicit form for the ring class field of the order $\mathbb{Z}[\sqrt{-19}]$ in the quadratic field $\mathbb{Q}(\sqrt{-19})$. Does anyone know if there is some and how it is?
1
vote
1answer
89 views

Isomorphism of the ideal class group with a cyclic group

Let $K=\mathbb{Q}(\sqrt{-17})$. Show that the ideal class group $Cl_K$ is isomorphic to $\mathbb{Z}/4\mathbb{Z}$. We know that the class number is 4...How i show that $Cl_K$ is cyclic?
6
votes
2answers
106 views

Hilbert class field application

If $K$ is an imaginary quadratic field and $M$ is an unramified Abelian extension of $K$, the prove that $M$ is Galois over $\mathbb{Q}$ Let see...If $L$ is the Hilbert class field of $K$, then $L$ ...
3
votes
2answers
147 views

Finding exercises in local fields, following Serre's book

I am reading Serre's "Local Fields". I would like to find more exercises to complement my study. So I am looking for, mabye, exercises from a course that followed this book. Do you know of such a ...
4
votes
1answer
118 views

What are the bounds on the class number of a cyclotomic field with regulator power of 2?

Let $\mathbb{Q}(\zeta_n)$ be the $n$th cyclotomic field with $n$ being a power of $2$. What is the best known asymptotic upper bound on the class number of $\mathbb{Q}(\zeta_n)$ as n grows? Can we ...
0
votes
0answers
149 views

Norm map of unramified extension

Let $K \subset L$ be an abelian unramified extension of local fields. Is it true that norm map $N:\mathcal O_L^*\mapsto \mathcal O_K^*$ is surjective?
4
votes
2answers
282 views

Class Group of $\mathbb Q(\sqrt{-35})$

As an exercise I am trying to compute the class group of $\mathbb Q(\sqrt{-35})$. We have $-35\equiv 1$ mod $4$, so the Minkowski bound is $\frac{4}{\pi}\frac12 \sqrt{35}<\frac23\cdot 6=4$. So we ...
1
vote
0answers
38 views

The criteria for two abelian extensions to be embedded

Learning class field theory I found this theorem, but I can't prove it or find the solution. I'll be glad to any help. Let $L$ and $M$ be abelian extensions of $K$. $L \subset M$ if and only if ...
5
votes
1answer
248 views

How to determine a Hilbert class field?

I tried to solve the exercise VIII.XX in Number Fields by Marcus. It asks to find the Hilbert class field of $Q(\sqrt m)$ for $m=-6,-10,-21,-30$. And the emphasis of this question is on the first two. ...
1
vote
0answers
39 views

How to measure the failure of Hasse norm theorem?

We know that the failure the unique factorisation is measured by the ideal class-group, that of the local-global principle depends upon the Tate-Shafarevich group. Then I thought: what should be ...
5
votes
3answers
144 views

Extension by adjoining roots of unity and ramification

Let $p$ be a prime, $n\geq 1$, $\zeta=\zeta_{p^n}$ a primitive $p^n$th root of unity, $L$ a number field, and $\wp$ a prime ideal of the ring of integers of $L$ lying above $p$. Suppose that ...
7
votes
0answers
187 views

Numbers represented by a cubic form

EDIT, April 11, 2013: See answer at http://mathoverflow.net/questions/127160/numbers-integrally-represented-by-a-ternary-cubic-form/127295#127295 This is part 2 ( of 25 discriminants of class number ...
12
votes
1answer
245 views

primes represented integrally by a homogeneous cubic form

Expired by this question Show determinant of matrix is non-zero I am moved to ask: Given integers $a,b,c,$ and cubic form $$ f(a,b,c) = a^3+2 b^3-6 a b c+4 c^3 = \left|\begin{bmatrix} a & 2c ...
14
votes
2answers
318 views

Galois Group of the Hilbert Class Field

Let $K/\mathbb{Q}$ be a number field with Galois group G and let $L/K$ be the Hilbert class field of $K.$ It is easy to show that $L$ is Galois over $\mathbb{Q}$ and I am interested in knowing this ...
4
votes
0answers
118 views

Complex multiplication - Ray class fields

I'm pretty new to complex multiplication and am struggling with Corollary 5.20 in Elliptic Curves with Complex Multiplication and the Conjecture of Birch and Swinnerton-Dyer by Rubin. According to ...
4
votes
0answers
39 views

Is the group of principal ideles closed in the group of finite ideles?

Let $K$ be an algebraic number field. Let $\mathbb{I}$ be the group of ideles and let $\mathbb{I}_f$ be the group of finite ideles. We embed $K^\times$ diagonally in both. It is know that $K^\times$ ...
15
votes
0answers
308 views

Which number fields can appear as subfields of a finite-dimensional division algebra over Q with center Q?

I have some idle questions about what's known about finite-dimensional division algebras over $\mathbb{Q}$ (thought of as "noncommutative number fields"). To keep the discussion focused, let's ...
12
votes
5answers
522 views

How does a Class group measure the failure of Unique factorization?

I have been stuck with a severe problem from last few days. I have developed some intuition for my-self in understanding the class group, but I lost the track of it in my brain. So I am now facing a ...
10
votes
0answers
151 views

CFT via Brauer groups vs via ideles

I am interested in the relationship between the following two versions of CFT: Version 1: (Brauer Group Version) Let $K$ be a number field. One constructs, for every finite place $v$ of $K$, a map ...
6
votes
0answers
120 views

Hecke characters and Unitary groups

Let $M/F$ be a quadratic extension of number fields, with Galois group $G=\{1,\tau\}$. Consider the following unitary group $$U_1(R)=\{z\in (R\otimes_FM)^\times :zz^\tau=1\},$$ where $R$ is an ...