All Questions
Tagged with cyclotomic-fields class-field-theory
10 questions
2
votes
0
answers
94
views
Is there a pure algebraic proof of the problem about class number of imaginary quadratic field $p\nmid h_{\mathbb{Q}(\sqrt{-p})}?$
For $p\equiv3\pmod4$, let $h_{\mathbb{Q}(\sqrt{-p})}$ denote the class number of $\mathbb{Q}(\sqrt{-p})$, the class number formula shows $h_{\mathbb{Q}(\sqrt{-p})} < p$. I wonder if there is a pure ...
- 21
5
votes
1
answer
478
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
161
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
2
votes
0
answers
108
views
Narrow class number of a the maximal totally real number field inside a cyclotomic field
I am wondering how much it is known about the narrow class number of the number field $\mathbb{Q}(\zeta_p+\zeta_p^{-1})$ ($p$ an odd prime). More precisely, I am interested to know when it is odd.
By ...
- 637
2
votes
0
answers
325
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
4
votes
0
answers
504
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
2
votes
1
answer
214
views
class group size of cyclotomic field subextension
In the following, let $\mathbb{Q_1}$ denote the subfield of degree $p$ over $\mathbb{Q}$ in the $p^2$- cyclotomic extension.
What is the best known upper bound for the size of its class group, $\text{...
- 250
10
votes
0
answers
358
views
Easy cases of Herbrand's theorem
$\def\QQ{\mathbb{Q}}\def\ZZ{\mathbb{Z}}$ I recall Herbrand's theorem about class groups of cyclotomic fields: Let $p$ be an odd prime, let $\zeta$ be a primitive $p$-th root of $1$ and let $K = \QQ(\...
- 156k
6
votes
2
answers
323
views
Computing the relative class group (with Galois action) of relatively large cyclotomic groups
For a cyclotomic field $K = \mathbb Q(\zeta_n)$, let $K^+$ be its maximal totally real subfield. We know that $H^+ = Cl(K^+)$ injects into $H = Cl(K)$. I am interested in computing the group $H/H^+$ ...
- 7,746
10
votes
1
answer
705
views
Tables of class numbers of cyclotomic fields
Does anyone have a table of the class numbers ($h_n$) of cyclotomic fields (upto say, n = 250-300 for $\mathbb Q(\mu_n)$)?
I can find tables for the relative class number ($h_n^-$) in various places ...
- 7,746