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
$$
\mathbb{A}_{K,f}^\times = \{(u_v)_v \in \prod_{v \nmid \infty} K_v^\times : v(u_v) = 0 \...
- 411
0
votes
1
answer
112
views
Ray class field and its conductor
Let $\mathfrak{m}$ be an ideal of the ring of integers of a number field $K$ and $S$ is a subset of the real places, then the ray class group of
$\mathfrak{m}$ and $S$ is the quotient group
$$I^{\...
- 287
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
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
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
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 $...
- 5,976
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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
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 $\...
- 5,976
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 $\...
- 5,976
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
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
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
-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 ...
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 ...
- 4,418
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
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
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
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 ...
7
votes
1
answer
343
views
Explicit cocycles for the first Galois cohomology of a $p$-adic torus
Let $K$ be a $p$-adic field (a finite extension of the field of $p$-adic numbers ${\mathbb Q}_p$).
Let $T$ be a $K$-torus with character group $X={\sf X}^*(T)$ and cocharacter group $Y={\sf X}_*(T)=X^\...
- 14.2k
2
votes
0
answers
125
views
Conditions for being an entry in a trace compatible sequence
$\DeclareMathOperator\Tr{Tr}$Let $K$ be a local field and let $q$ be the size of the residue field of $K$. $\pi$ will be a uniformizer of $K$. Let $f(X) = \pi X + X^q$. Then there is a unique formal ...
- 123
4
votes
1
answer
322
views
How to calculate genus number of number field using sage?
I am looking to find real quadratic fields whose Hilbert class field is abelian over $\Bbb Q$. Then I learned about genus numbers and genus field of the number field. It is enough to find a number ...
- 753
2
votes
0
answers
110
views
Is there data base of quadratic fields which have abelian Hilbert class field?
I did not find it in LMFDB. However, I am looking for a Database; please feel free to navigate me to any paper, which might be helpful.
I really appreciate any help you can provide.
- 343
4
votes
0
answers
305
views
Explicit invariant map in local class field theory
Let $K$ be a $p$-adic field with algebraic closure $\overline{K}$. Then if $K^\text{nr}$ is the maximal unramified extension of $K$ in $\overline{K}$, there is an explicit invariant map:
$$
H^2(\...
- 953
2
votes
0
answers
153
views
Construction of genus class fields
Given a finite extension $K/\mathbb{Q}$, the genus class field $L$ is defined to be the maximal abelian extension of $\mathbb{Q}$ that is a subfield of the Hilbert class field $H$ of $K$. I am trying ...
- 577
3
votes
1
answer
198
views
Centralizer of the absolute Galois group of a number field
By this answer, we know that if $K/\mathbb{Q}_p$ is a finite extension, the centralizer of $G_K$ in $G_{\mathbb{Q}_p}$ is trivial. The argument there uses that the abelinization of $G_K$ is the pro-...
- 4,418
3
votes
0
answers
166
views
Are class numbers of number fields with prime degree often $1$?
I have taken a look at the class number statistics of the L-functions and Modular Forms Database:
https://www.lmfdb.org/NumberField/stats, table "Distribution by class number".
It appears ...
- 125
2
votes
1
answer
154
views
Are the maximal cyclotomic field contained in a number field and its Hilbert class group the same?
Let $K$ be a number field. If $d$ be the smallest even integer such that $\Bbb Q (\zeta_d) \subset K,$ then I wanted to prove that if $d'>d$ then $\Bbb Q (\zeta_{d'}) \not\subset H(K),$ where $H(...
- 753
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
2
votes
1
answer
184
views
Does K and its Hilbert class field have same conductor?
Let $K$ be an abelian number field and $H(K)$ be the Hilbert class field of $K.$
Definition: (conductor of a abelian number field) Let $K$ be a number field with the abelian Galois group over $\Bbb{Q}....
- 343
2
votes
0
answers
327
views
Definition of Euler system of cyclotomic units
I am not sure about my understanding of Euler system of cyclotomic unit. This is what I have learnt:
Let $F=\mathbb{Q}(\mu_m)$.
Let $\mathcal{I}(m)$ = {positive square free integers divisible only by ...
- 99