All Questions
Tagged with cyclotomic-fields nt.number-theory
57 questions
3
votes
0
answers
112
views
The maximal $p$-abelian $p$-ramified extension of the cyclotomic $\mathbb{Z}_p$-extension
$\DeclareMathOperator\Gal{Gal}\DeclareMathOperator\Cl{Cl}$[Reference: S. Lang, Cyclotomic Fields I and II, §2 chap 6]
Let $K$ be a number field. Suppose that $K$ contains the $p$-th roots of unity if $...
- 367
6
votes
0
answers
252
views
Abelian extensions of Q and cyclotomic fields
I have changed some notation based on the comments of Chris Wuthrich and Wojowu.
For an abelian extension $F$ of $\mathbb{Q}$, let $c(F)$ be its conductor. That is, $c(F)$ is the smallest positive ...
- 385
5
votes
1
answer
355
views
Diophantine equation $\cos(2\pi x)\cos(2\pi y) = \cos(2\pi z)$
While working on finite order elements of $\operatorname{SO}_n$, I meet this question:
Find all identities of the form $\cos(2\pi x)\cos(2\pi y) = \cos(2\pi z)$ with $x, y, z$ rational numbers.
As ...
- 3,432
6
votes
0
answers
136
views
Is the minus class group isomorphic to the relative class group?
I think this is something I should have known, but if I ever did I forgot about it. Consider the field $L$ of $p$-th roots of unity ($p$ prime) and its maximal real subfield $L^+$. The transfer of ...
- 32.5k
4
votes
0
answers
94
views
How to describe a concrete generator of $\widetilde{K_0(\mathbb{Z}[C_{23}])} \cong \widetilde{K_0(\mathbb{Z}[\zeta_{23}])}$
Milnor (page 29, see below) gives an explicit proof that the zeroth $K$-theory of the group ring $\mathbb{Z}[C_p]$, where $C_p$ is the cyclic group of order $p$ with $p$ a prime agrees with $K_0$ of $\...
- 2,303
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
3
votes
0
answers
226
views
Generalisation of Sharifi's conjecture for Siegel varieties
I recently learned Sharifi's conjecture in this article by F.Takako and K.Kato.
According to my understanding, this conjecture states that (the conjugation invariant) of the projective limit $\...
- 1,270
0
votes
0
answers
85
views
Relation between two finite abelian extensions of rationals
For $\mathbb{F}$ a finite abelian extension of $\mathbb{Q}$, recall that the conductor $c(\mathbb{F})$ of $\mathbb{F}$ is the smallest positive integer such that $\mathbb{F}$ is contained in the $c(\...
- 385
3
votes
0
answers
135
views
Recover cyclotomic integer with bounded coefficients from its known associate
Recall that two cyclotomic integers are called associated if their ratio is a unit in the corresponding ring of cyclotomic integers.
We will view cyclotomic integers as polynomials (of degree $<\...
- 34.3k
5
votes
1
answer
931
views
The group of all units of integral cyclotomic ring
Let $\zeta_n = e^{i2\pi/n}$. What is the group of all units in the integral cyclotomic ring $\mathbb{Z}[\zeta_n]$?
Here I like to know all the group elements for small $n$'s. For $n=1$ and $n=2$, the ...
- 4,826
7
votes
0
answers
149
views
Finding when a certain product in a cyclotomic field is equal to one
For the following, fix $m\geq 2$, and let $a_{0},\dots,a_{m-1}\in\mathbb{Z}$ be such that $\sum_{k=0}^{m-1}a_{k}=0$. I would like to find the exact conditions on the $a_{k}$ so that the following ...
- 299
4
votes
2
answers
574
views
Existence of solution for a system of quadratic diophantine equations / symmetric quadratic froms
I am interested in solving, or even just deciding the existence of a solution, for a system of quadratic diophantine equations.
Let $p$ be a prime congruent to 1 modulo 8, so $ p =17$ is the first ...
- 91
1
vote
0
answers
85
views
How to describe the automorphisms of $\mathbb{Q}(\theta)$ that fix $\mathbb{Q}(\sqrt{-m})$
Suppose $m > 0$ is a square free integer and $m \equiv 1 $mod $4$. Then the $K = \mathbb{Q}(\sqrt{-m})$ is a subfield of $L = \mathbb{Q}(\theta)$, where $\theta$ is a primitive $4m$-th root of ...
- 707
2
votes
0
answers
84
views
Question about infinitude of $m$-irregular primes
Let $p>2$ be a prime, $\zeta$ a primitive $p$th root of unity, and $m >0$ a square-free integer such that $m \not\equiv 3 \mod 4$ and $\gcd(m,p)=1$. Let $\chi$ be the imaginary quadratic ...
- 707
0
votes
1
answer
127
views
The average number of solutions to $a+b=c$ in a multiplicative subgroup of $\mathbb{F}_q^\times$ when $c\in\mathbb{F}_q^\times$ is random
$\mathbb{F}_q^\times$ is the multiplicative group of the finite field $\mathbb{F}_q$, and H is a multiplicative subgroup of $\mathbb{F}_q^\times$ of order $r<q−1$.
What is the average number of ...
1
vote
0
answers
103
views
"multi-dimensional" cyclotomic number
Let $F_q$ be the finite field with $q$ elements with characteristic $p$ and with $g$ being a primitive root. Let $N$ be a divisor of $q-1$ and let $C_0$ be the subgroup of $F_q^*$ with index $N$. Then ...
- 91
1
vote
1
answer
393
views
Classification of cyclotomic fields with class number 1
1.Let $p$ be an odd prime number. Is there a classification of cyclotomic fields of the form $\mathbb{Q}(\zeta_p)$ with class number 1?
2.Is there such a classification for general cyclotomic fields $...
- 333
6
votes
0
answers
500
views
Galois cohomology with coefficients in the unit group of a cyclotomic field
While understanding Fermat's last theorem's proof for regular primes, I bumped into a proposition (http://www2.biglobe.ne.jp/~optimist/algebra/fermat2_proof.html#proof10, written in Japanese) stating:
...
- 369
3
votes
1
answer
270
views
Connecting different ways of constructing cubic extensions of $\mathbb{Q}$
There are at least two ways to construct cyclic cubic extensions of $\mathbb{Q}$ as explained below. (A third one is given in the answer to an earlier question).
Given $A, B, C$ integers with $A\neq ...
- 1,566
24
votes
1
answer
860
views
Why is the regulator of the degree 11 extension of $\mathbb{Q}$ so large?
Let $K_\infty/\mathbb{Q}$ denote the $\hat{\mathbb{Z}}$-extension of $\mathbb{Q}$. Then for each $n\geq1$, $K_\infty$ has a unique subfield $K_n$ of degree $n$ over $\mathbb{Q}$.
The fields $K_n$ are ...
- 2,869
4
votes
1
answer
205
views
Multiplicative set of positive algebraic integers
Let $S$ be a set of algebraic integers such that:
$\mathbb{N}_{\ge 1} \subseteq S \subset \mathbb{R}_{\ge 1}$,
$\alpha, \beta \in S \Rightarrow \alpha \beta \in S$,
$\alpha, \beta \in S \Rightarrow ...
9
votes
1
answer
235
views
Are the class numbers of $\mathbb{Q}(\cos(2\pi / m))$ $O(m^n)$ for some fixed $n$?
Question: Are the class numbers of $\mathbb{Q}(\cos(\frac{2\pi}m))$ $O(m^n)$ for some fixed $n$?
Evidences (e.g. a recent paper) showing that the question above is open are also OK.
Remark: If such $n$...
- 9,501
12
votes
0
answers
567
views
Is class group of cyclotomic fields cyclic?
What are the cyclotomics fields with a cyclic class group. I read that there are only 29 cyclotomic extensions of $\Bbb Q$ with class number one. But I wanted to know what condition on $n$ would make $...
- 743
1
vote
0
answers
109
views
Euler systems over abelian number fields [duplicate]
Im confused with the following statement:
Coleman’s conjecture concerning circular distribution imply that Euler systems over abelian number fields arise in “an elementary” way from the theory of ...
- 99
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
6
votes
0
answers
126
views
Simultaneous vanishing $\mathbb{Q}$-linear relations between $N$-th roots of unity
Let $\zeta$ be a primitive $N$-th root of unity and $\Gamma \subset (\mathbb{Z}/N)^\times$ a subgroup. Let $|\Gamma|$ be the cardinality of $\Gamma$ and consider the linear map $M_\Gamma\colon \mathbb{...
- 171
10
votes
0
answers
561
views
Toward a cyclotomic Riemann hypothesis
For an integer $n \ge 3$, consider the function $$u(n) = \frac{\sigma(n)}{n \log \log n}$$ with $\sigma$ the divisor function. Now consider the sequence (bounded below and decreasing) $$v_n = \sup_{m&...
4
votes
1
answer
376
views
Quadratic extensions of cyclotomic numbers by absolute values of elements
Summary
I was wondering whether there is an explicit description of the extension $\mathbb{Q}(\zeta_n, |z|)/\mathbb{Q}$ obtained from a cyclotomic field $\mathbb{Q}(\zeta_n)$ by adjoining any finite ...
- 627
3
votes
0
answers
110
views
Minimum of a product of polynomial evaluated at primitive roots of unity, given that the value of the polynomial at the same lies on unit circle
This is something that came out of working on a problem:
Let $m$ be an odd positive integer and $f \in \mathbb Q[x]$ be a polynomial of degree less than $m$. With $\zeta_m$ denoting a primitive root ...
- 739
1
vote
0
answers
231
views
Regulator of number fields of a special form
Is it possible to estimate the regulator of a number field of a special form (e.g. cyclotomic fields)?
My motivation mainly comes from the investigation of the number fields $K_n=\mathbb{Q}[x]/(x^n+x+...
- 9,501
9
votes
1
answer
380
views
Is an integral sum of periodic vectors always a sum of integral periodic vectors?
Update:
I have found reference to this problem. It is known as "the Rédei-de Bruijn-Schönberg theorem", which is proved in the following papers:
N. G. de Bruijn: On the factorization of cyclic ...
- 3,432
1
vote
1
answer
299
views
How to find a cyclotomic polynomial of degree d that decompose into d irreducible polynomials in $Z_6$?
More specifically, I need the degree $d$ to be around 1024. I can easily find the cyclotomic polynomial of degree 1024 that satisfies the above requirement in $Z_2$, i.e., $x^{1024}+1$, which is equal ...
- 33
10
votes
1
answer
466
views
Is there a pattern for the irreducible factors and degrees of a characteristic polynomial?
Let $Log(n) = \sum_{i=1}^r \alpha_i \cdot e_i$, where $n = \prod_{i=1}^r p_i^{\alpha_i}$ and $p_i$ is the $i$-th prime, $\alpha_i \ge 0$, $e_i$ is the $i$-th standard basis vector. For example $6 = 2\...
user6671
28
votes
2
answers
2k
views
A sum involving roots of unity
Let $n$ be a positive integer and $\zeta$ be a primitive $n$th root of unity. It is not hard to show that
\begin{align*}
\sum_{k=1}^{n-1}\frac{\zeta^k}{1-\zeta^k}=\frac{1-n}{2}.
\end{align*}
Since $\...
- 2,183
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
1
answer
948
views
Splitting of primes in cyclotomic $\mathbb{Z}_p$-extension
Let $p$ be a prime, $K$ be a finite extension of $\mathbb{Q}$ and $K_{\infty}$ be a cyclotomic $\mathbb{Z}_p$-extension of $K$ i.e. Gal$(K_{\infty}/K) \cong \mathbb{Z}_p$, the group of $p$-adic ...
- 193
19
votes
3
answers
1k
views
Points of elliptic curves over cyclotomic extensions
Let $E$ be an elliptic curve over $\mathbb Q$. Let's look at the group of points of this elliptic curve over $\mathbb Q(1^{1/\infty})$ which we get after adding all roots of unity to $\mathbb Q$. It ...
- 2,305
2
votes
0
answers
218
views
Cyclotomic ring of integers proof via matrix theory
Not sure of a better title (and I'm open to suggestions). But when following Laffey's notes on Integer Matrices, found here (and in particular, starting on page 13), I came across an alternative proof ...
1
vote
0
answers
191
views
On discrepancy properties of $\{0,\pm1\}$ sequences arising from cyclotomic polynomials
Given $n=2^k p^r q^m$ take a $d\in\Bbb Z$ with $d\mid n$ such that each $a_i$ is in $\{0,\pm1\}$ in $$f_{d,n}(x)=\frac{x^n-1}{\Phi_d(x)}=a_0+a_1x+\cdots+a_{\deg(f_{d,n}(x))}x^{\deg(f_{d,n}(x))}.$$
...
- 13.9k
1
vote
1
answer
201
views
On largest degree of polynomial related to cyclotomic polynomials - I
We know cyclotomic polynomials $\Phi_{2^kp^rq^m}(x)$ have coefficients in $\{0,\pm1\}$.
What is the largest degree $f_{d,n}(x)=\frac{x^{2^{k}p^{r}q^{m}}-1}{\Phi_d(x)}$ with $\{0,\pm1\}$ coefficients ...
- 13.9k
7
votes
2
answers
291
views
Elements of absolute value 1 in cyclotomic extension of dyadic rationals
Let $\zeta$ be a primitive $2^k$th root of unity, and consider the ring $ \mathbb{Z}[\zeta, \frac{1}{2}] \subset \mathbb{C}$. Are there any elements of absolute value 1 other than the powers of $\zeta$...
- 12.4k
4
votes
0
answers
422
views
Cyclotomic Extension of a Perfectoid Space
Maybe, I am being stupid, but when I consider ramified extension of a perfectoid field with the characteristic $0$, I cannot find the correspondent field with characteristic $p$. Let me put it more ...
- 41
7
votes
3
answers
771
views
Something interesting about the quintic $x^5 + x^4 - 4 x^3 - 3 x^2 + 3 x + 1=0$ and its cousins
(Update):
Courtesy of Myerson's and Elkies' answers, we find a second cyclic quintic for $\cos\frac{2\pi}{p}$ with $p=\text{1 mod 10}$ as,
$$\frac{z^5}{\beta} = 10 z^3 - 20 n^2 z^2 + 5 (3 n^4 - 25 n^...
- 12.6k
4
votes
0
answers
158
views
Unique factorization for the semigroup generated by {2cos(π/n) | n>3}?
Let $S$ be the multiplicative semigroup of numbers generated by $B=\{ 2cos(\frac{\pi}{n}) \mid n \ge 4 \}$.
Question: Does every number of $S$ factorize uniquely (up to perm.) as a product of ...
4
votes
1
answer
529
views
Unit in cyclotomic field
Let $n \in \mathbb{N}$ and $\zeta$ be a primitive $n$-th root of unity. I want to know for which $n$ the element $1+2(\zeta+\zeta^{-1})$ is a unit in the ring of integers of $\mathbb{Q}[\zeta]$. Can ...
- 41
5
votes
1
answer
463
views
Motivation for cyclotomic units
I am wondering the original motivation for considering cyclotomic units. Maybe one can rephrase the question as:
Why did people initially consider such units in $\mathbb{Q}(\zeta_p)$ specially?
...
- 1,428
5
votes
1
answer
409
views
A strange condition on containment of special complex numbers in cyclotomic fields
In a recent theorem we have naturally come across this condition, that seems to be important, but rarely satisfied:
$\sqrt{\frac 1 4 + a^m} \in \mathbb Q(\zeta_m, a)$
where $a\in\mathbb C^*$ and $\...
18
votes
1
answer
864
views
What are the products $\prod_{A\subset{\mathbb F}_p\colon |A|=n} \sum_{a\in A} \zeta^a$ equal to?
This is a somewhat more explicit version of a question I have recently asked.
Let $p$ be an odd prime, and write $\zeta:=\exp(2\pi i/p)$ (any other primitive $p$th root of unity will do as well). For ...
- 23k