Class field theory is a major branch of algebraic number theory that studies abelian extensions of global fields.
2
votes
2answers
55 views
On the Pell-like $px^2-qy^2 = 1$ for prime $p,q$
Given any prime of form $p_n = u^2+nv^2$ for non-zero integers $u,v$. Consider,
\begin{aligned}
&p_2x^2-2y^2 = 1\\
&p_3x^2-3y^2 = 1\\
&p_7x^2-7y^2 = 1\\
&p_{11}x^2-11y^2 = 1\\
...
4
votes
2answers
91 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
26 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 ...
3
votes
1answer
40 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. ...
4
votes
0answers
72 views
$S$-Units notation and Dirichlet's unit theorem
I'm having a hard time understanding some notions of a paper I'm working on. Let $L/K$ be a finite normal extension of number fields and $S$ be a set of places of $K$ prime to $p$ where $p$ denotes an ...
4
votes
0answers
44 views
Non-number-theoretic class field theory?
This is a curiosity-oriented question, so references or indications are very welcomed.
In the book on class field theory by Neukirch, one finds an abstract version of class-field theory: for a ...
0
votes
0answers
18 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
63 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
67 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 ...
10
votes
1answer
109 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 ...
12
votes
2answers
160 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
66 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 ...
5
votes
0answers
88 views
What numbers are integrally represented by $4 x^2 + 2 x y + 7 y^2 - z^3$
EDIT: a selection of material relevant to this problem is available at INHOMOGENEOUS
In December 2010 my question appeared in the M.A.A. Monthly, show that $4 x^2 + 2 x y + 7 y^2 - z^3 \neq \pm 2 ...
3
votes
0answers
32 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$ ...
2
votes
0answers
90 views
Base-Change for $GL_1$
I try to understand the base-change in the theory of automorphic forms for the simplest case: $GL_1$.
Let $L/K$ be an abelian extension of number fields of degree $n$ and let $\Gamma$ be the set of ...
13
votes
0answers
190 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 ...
9
votes
5answers
329 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 ...
7
votes
0answers
109 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
99 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 ...
