7
votes
1answer
104 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
27 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
44 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
30 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
62 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
52 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
41 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
144 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
50 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
54 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
102 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
99 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
29 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
50 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
102 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
168 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
264 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
88 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
146 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
279 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 ...
4
votes
1answer
242 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
143 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
186 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
240 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
317 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
116 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
306 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
516 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
119 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 ...