All Questions
144 questions
3
votes
1
answer
123
views
Reference request: ray class group as quotient of finite ideles
Let $K$ be a number field, and write $\mathbb{A}_{K,f}^\times$ for the group of finite ideles of $K$. That is
$$
\mathbb{A}_{K,f}^\times = \{(u_v)_v \in \prod_{v \nmid \infty} K_v^\times : v(u_v) = 0 \...
- 411
3
votes
3
answers
388
views
On subfields of the cyclotomic field $\mathbb{Q}(\zeta_p)$
Let $p$ be an odd prime. Let $\zeta_p=e^{2\pi{\bf i}/p}$ and let $1\le k\le p-1$ be a divisor of $p-1$. Recently, when I learnt algebraic number theory, I met the following problem.
If we let
$$U_k=\{...
- 87
0
votes
1
answer
112
views
Ray class field and its conductor
Let $\mathfrak{m}$ be an ideal of the ring of integers of a number field $K$ and $S$ is a subset of the real places, then the ray class group of
$\mathfrak{m}$ and $S$ is the quotient group
$$I^{\...
- 287
2
votes
1
answer
175
views
Relation between the genus number and the ambiguous class number
It is well known that for $K/F$ a finite "cyclic" extension of number fields, we have $g_{K/F}=a_{K/F}$, where $g_{K/F}$ denotes the relative genus number, and $a_{K/F}$ denotes the ...
- 437
2
votes
1
answer
128
views
Conductor of the Hecke character- power residue symbol
The power residue symbol is the multiplicative character $\bigl(\frac a \cdot\bigr): \mathcal I_\mathfrak m\to \mu_n$ that satistfies $\bigl(\frac a {\mathfrak p}\bigr)\equiv a^{\frac{N\mathfrak p-1}{...
- 133
2
votes
0
answers
116
views
Restriction of the local Artin map on the valuation ring of a local field
Let $F$ be a local field, in particular a finite extension of $\mathbb{Q}_p$ for some prime $p$ and let $Art_{L}: L^\times \to Gal(F^{ab}/F)$ be its local Artin map. We know that if $L/F$ is a finite ...
- 367
2
votes
1
answer
88
views
Cyclic extensions of a number field of full local degree in a given set $S$
Let $K$ be a number field, and let $S=S_f\cup S_{\mathbb R}\cup S_{\mathbb C}$
be a finite set of places of $K$, where $S_f$ denotes the set of finite places in $S$, $S_{\mathbb R}$ denotes the set of ...
- 14.2k
4
votes
1
answer
224
views
Generators of the ideal class group
Theorem 4 of Eric Bach's "Explicit bounds for primality testing and related problems" states the following:
Let $K$ be a number field of degree greater than 1. Let $d$ be the absolute value ...
- 43
4
votes
1
answer
190
views
Class numbers in the unramified biquadratic extensions of number fields
Let $K/k$ be an unramified biquadratic extension of number fields (i.e., $\operatorname{Gal}(K/k)\simeq V_4$), and $k_1$, $k_2$ and $k_3$ its three intermediate fields. I know, in general, we can ...
- 65
4
votes
1
answer
367
views
Conductor and local Kronecker–Weber theorem
Given an abelian extension $K$ of $\mathbb{Q}$, the global Kronecker–Weber theorem tells us that there exist a positive integer $N$ and a primitive $N$-th root of unity $\zeta_N$ such that $K\subseteq ...
- 549
1
vote
1
answer
140
views
Defect between modulus and conductor of ray class field
I have following question about a remark in J. Neukirch's
Algebraic Number Theory around page 397.
The context: We consider ideal theoretic formulation of global class field theory of a number field $...
- 5,986
2
votes
2
answers
433
views
Expositions of the classical approach to local class field theory (Brauer group and Hasse invariant)
I've posted this question already on MSE and didn't get much out of it, so I hope it's OK to repost here.
I'm an undergraduate trying to learn local class field theory from the corresponding chapter ...
- 121
18
votes
4
answers
2k
views
What's the use of group cohomology for class field theory?
I'm a graduate student studying now for the first time class field theory.
It seems that how to teach class field theory is a problem over which many have already written on MathOverflow.
For example ...
- 531
2
votes
0
answers
65
views
Constructing a cyclic extension $L$ with given local behavior of a global field $K$ such that $L$ is normal over a subfield $F$ of $K$
Let $F$ be a global field without real places
(that is, a function field or a totally imaginary number field).
Let $K/F$ be a cyclic extension of degree $n$.
Let $S$ be a ${\rm Gal}(K/F)$-invariant ...
- 14.2k
1
vote
0
answers
58
views
Normality in a tower of cyclic extensions of global fields, as in Artin-Tate
Let $L_0$ be a global field without real places, that is, a global function field or a totally imaginary number field,
and let $V_f(L_0)$ denote the set of finite (that is, non-archimedean) places of $...
- 14.2k
7
votes
0
answers
157
views
Non-abelian ray class fields for local fields
Let $K$ be a non-Archimedean local field. Then, thanks to work of Koch (when $K$ has positive characteristic) and Jannsen-Wingberg (when $K$ has characteristic zero, and odd residual characteristic) ...
- 1,116
4
votes
0
answers
147
views
A normal extension of a number field of given degree that does not split over a given set of finite places
Let $K$ be a number field and $S$ be a finite set of non-archimedean places of $K$. Let $n>1$ be a natural number.
Question. Does there exist a normal extension $L/K$ of degree $n$ such that $L\...
- 14.2k
2
votes
0
answers
250
views
Maximal p-extension and pro-p extension
I’m studying Iwasawa theory and I meet some questions Thanks a lot for your help.
Q_1: About terminology $p$-extension.
I find many reference use maximal $p$-extension or maximal abelian p-extension ...
- 319
64
votes
3
answers
5k
views
Class field theory - a "dead end"?
I found the claim in the title a bit astonishing when I first read it recently in an interview with Michael Rapoport in the German magazine Spiegel (8 February 2019). And I was wondering how he comes ...
- 2,810
6
votes
1
answer
427
views
Surjectivity of the norm of units in Galois extensions ramified exactly at one finite prime
Let $L/K$ be a finite Galois extension of number fields that is ramified exactly at one finite prime and is unramified at all infinite primes. Let $U_K$ and $U_L$ denote the units of the ring of ...
- 333
8
votes
2
answers
479
views
Image of the norm map for Artin-Schreier extensions
Let $k$ be a local field of characteristic $p$ and $\omega \in k$ a uniformiser. Consider the Artin-Schreier extension $L_n = k[x]/(x^p - x - \omega^n)$ for each $n \in \mathbb{Z}$.
Is there an ...
- 21.4k
4
votes
0
answers
181
views
The order of the global Galois group
For any profinite group $G$, we can define the order of $G$ using the notion of supernatural number. Now let $K$ be a number field, $S$ a finite set of primes of $K$ and $ G_{K,S} $ the Galois group ...
- 863
5
votes
0
answers
176
views
Epstein zeta function for non-fundamental discriminant to L-series
Let $Q(x,y) = ax^2+b xy + cy^2$ be a primitive integral positive-definite quadratic form, with associated number field $K$. If $D=b^2-4ac$ is a fundamental discriminant, then it's well-known that $$\...
- 528
5
votes
1
answer
485
views
Class numbers of cyclotomic fields and their maximal totally real subfields
Let $\zeta_p$ be a $p$-th root of unity for a prime $p$, let $L:=\mathbb{Q}(\zeta_p)$ and $K$ the maximal totally real subfield of $L$, i.e. $K:=\mathbb{Q}(\zeta_p+\zeta_p^{-1})$. I am trying to prove ...
- 637
1
vote
0
answers
100
views
Examples of $\mathbb{Z}_p$-extensions and two $\mathbb{Z}_p$-extensions with a "nontrivial" intersection
Let $k$ be a number field (a finite extension of $\mathbb{Q}$). Let $p$ be a prime. By saying "$\mathbb{Z}_p$-extensions", we mean Galois extensions $K/k$ of Galois group isomorphic to $\...
- 639
50
votes
13
answers
6k
views
Erratum for Cassels-Froehlich
Edit 25 April 2010: I have a physical copy of the new printing of the book. I can only assume the LMS is now selling it (but have no details).
IMPORTANT EDIT: THE RESULTS ARE IN! Ok, the deadline has ...
2
votes
0
answers
93
views
Compositum of field extensions in context of $\mathbb Z_p$ extension
I had asked this question on stackexchange and I think it is better suited for this site.
Suppose I have a $\Gamma \simeq \mathbb Z_p $ extension $F_\infty /F$ of a number field $F$. Let $F_n$ be the ...
- 121
1
vote
2
answers
169
views
The $ 1 \operatorname{ mod } \mathfrak{m}$ congruence relation in ray $ P^{\mathfrak{m}}$ of the ideal theoretic ray class group
Let $K$ be a number field, $\mathcal{O}_K$ its ring of integers and $S$ a
subset of the real places. Let $\mathfrak{m} \subset \mathcal{O}_K$ an ideal. The ideal
theoretic
ray class group of $\...
- 5,986
3
votes
1
answer
176
views
Knot group of a field extension
Notation:
$L/K$, finite extension of global fields
$K^\times$, unit group of $K$
$L^\times$, units group of $L$
$\mathbb{A}_L^\times$, ideles of $L$
$N_{L/K}$, the norm map
The knot group of an ...
- 595
2
votes
1
answer
307
views
A question about unramified quadratic extension of number field
Is there any condition over a number field $K$ for an unramified quadratic extension of $K$ to admit an embedding into an unramified cyclic extension of degree 4 of $K$?
- 65
-1
votes
1
answer
180
views
Set of all primes $p$ that split in $\mathbb{Q}\left(\sqrt{-k}\right)$
Let $k$ be a squarefree positive integer. We know that a prime $p$ splits in $K=\mathbb{Q}(\sqrt{-k})$ if and only if $-k$ is a quadratic residue mod $p$.
My question is: can we explicitly determine ...
0
votes
0
answers
200
views
What is the conductor of $K(\sqrt{2})$ over $K$?
Let $ K=\Bbb Q\left(\sqrt{(-1)^\frac{N+1}{2}N}\right)$. I want to find the ray class field of $K$ containing $\sqrt{2}$. I considered $L=\Bbb Q(\sqrt{2})$. By Artin reciprocity, there exist modulus $\...
- 753
4
votes
1
answer
167
views
Existence of lift of (local) Artin map
In a comment to this question, David Loeffler asked if one can show that the (local) Artin map
$$K^\times \to G_K^{ab}$$
does not have a lift to $G_K$. Probably this wouldn't be canonical, but I can't ...
- 4,428
3
votes
0
answers
162
views
relating class number and narrow class number of a real field
I am interested in finding out when the narrow class number of $\mathbb{Q}(\zeta_p+\zeta_p^{-1})$ is the same as the class number of $\mathbb{Q}(\zeta_p+\zeta_p^{-1})$ where $\zeta_p$ is a primitive $...
- 637
73
votes
2
answers
10k
views
Please check my 6-line proof of Fermat's Last Theorem.
Kidding, kidding. But I do have a question about an $n$-line outline of a proof of the first case of FLT, with $n$ relatively small.
Here's a result of Eichler (remark after Theorem 6.23 in ...
- 8,467
3
votes
1
answer
357
views
Lang's proof concerning ray class fields of imaginary quadratic number fields
Crosspost from Math.SE as I did not receive an answer there:
In Lang's book Elliptic Functions, he shows how to generate the ray class field with conductor $N$ of an imaginary quadratic number field $...
- 199
34
votes
4
answers
3k
views
$A_5$-extension of number fields unramified everywhere
So I was having tea with a colleague immensely more talented than myself and we were discussing his teaching algebraic number theory. He told me that he had given a few examples of abelian and ...
- 10.9k
9
votes
2
answers
902
views
How to compute with the Stark conjectures?
I would like a convenient basis for the elements of a fixed abelian extension $E$ of a real quadratic field $\mathbb{Q}(\sqrt{d})$. The accepted answer to this MO question suggests that the Stark ...
- 7,633
9
votes
1
answer
827
views
how to compute Hilbert class field of $\Bbb Q(\zeta_{23})$?
I want to construct the Hilbert class field of $K=\Bbb Q(\zeta_{23}).$ I have no clue how to construct it except that I know that $[H(K):K]=3$ from Sage. Any references or comments are appreciated.
- 343
4
votes
0
answers
506
views
Euler Systems and Coleman’s Conjecture
I’m trying to work on Coleman’s conjecture for abelian extensions of imaginary quadratic fields. I’ve read most papers by Seo regarding circular distributions. However, I’m a still confused about what ...
- 99
1
vote
0
answers
116
views
CM-fields and ideal theoretic ray class group
Let $K$ be a number field, $\mathcal{O}_K$ its ring of integers
and $S$ a finite subset of the real places. Let $\mathfrak{m}
\subset \mathcal{O}_K$ an ideal. The ideal
theoretic
ray class group of $\...
- 5,986
42
votes
2
answers
5k
views
Motivating Lubin-Tate theory
The Lubin-Tate theory gives an amazingly clean and streamlined way of constructing the subfield (usually denoted) $F_\pi\subset F^\mathrm{ab}$ for a local field $F$ fixed by the Artin map associated ...
- 910
4
votes
0
answers
164
views
What are the applications of $\lambda$-rings to class field theory?
In the book Lambda Rings by Yau, he mentions several areas where $\lambda$-rings can be applied, but he doesn't go into much details. He even includes class field theory in the list, mentioning "...
- 804
1
vote
0
answers
114
views
Rayclass group and Hilbert class group, $\mathrm{Gal}(K(\mathfrak{a})/K)\cong(\mathcal{O}_K/\mathfrak{a})^\times$
Let $K$ be an imaginary quadratic field with class number $1$. Let $\mathfrak{a}$ be an ideal of $ \mathcal{O}_K$ and $K(\mathfrak{a})$ denote the ray class field of $K$ modulo $\mathfrak{a}$.
Why ...
- 1,541
26
votes
4
answers
3k
views
Why do congruence conditions not suffice to determine which primes split in non-abelian extensions?
How does one prove that the splitting of primes in a non-abelian extension of number fields is not determined by congruence conditions?
user19475
7
votes
0
answers
205
views
Reference request: projectivity of the absolute Galois group of $\mathbb{Q}^{\mathrm{ab}}$
$\newcommand{\ab}{\mathrm{ab}}$Let $\mathbb{Q}^{\ab}$ denote the maximal abelian extension of $\mathbb{Q}$. I have heard the absolute Galois group of $\mathbb{Q}^{\ab}$ is projective (e.g. see this ...
1
vote
0
answers
200
views
Units in residue classes modulo prime ideal
Let $K$ be a number field and $\mathcal{O}_K$ be its ring of integers with unit rank $\geq 1$. For a given prime ideal $\mathfrak{p}$, shall we say that the map $\mathcal{O}_K^\times \to (\mathcal{O}...
- 11
4
votes
1
answer
400
views
Dihedral extension of $\mathbb Q$ with small discriminant
Let $K$ be a fixed quadratic number field, say $K=\mathbb Q(\sqrt 5)$. For any integer $n \geq 3,$ I would like to build a number field $D_n$ such that $D_n/\mathbb Q$ is Galois, with Galois group ...
- 1,322
12
votes
2
answers
605
views
Conceptual explanations of the class numbers for the first few $\mathbb{Q}(\sqrt{p})$ with odd conductor
It's known that the class number of $\mathbb{Q}(\sqrt{p})$ is $1$ for all primes $p<229$.
Question: What would it be like for conceptual explanations of $h(\mathbb{Q}(\sqrt{p}))=1$ for the first ...
- 9,501
25
votes
2
answers
4k
views
Primes of the form $x^2+ny^2$ and congruences.
The answer of following classical problem is surely known, but I can't find a reference
For which positive integer $n$ is the set $S_n$ of primes of the form $x^2+n y^2$ ($x$, $y$ integers) ...
- 26.1k