Questions tagged [cyclotomic-fields]

A cyclotomic field is a number field obtained by adjoining a complex primitive root of unity to $\mathbb Q$, the field of rational numbers.

Filter by
Sorted by
Tagged with
0 votes
0 answers
45 views

frobenius map on primitive nth root of unity over Fp(w) with (n,p)=1

Suppose w is a primitive n-th root of unity, let r = Ord_n(p) (the smallest integer s.t. $p^r=1 \pmod n $), $f(w)=w^p$(the ...
user avatar
3 votes
0 answers
60 views

Minimal Norm Vectors in certain Cyclotomic Ideal Lattices

Let $q$ be an odd prime and let $\zeta$ be a primitive $q^{th}$ root of unity. Let $I$ be the ideal in $\mathbb{Z}[\zeta]$ generated by $1-\zeta$. It is known that we can give $I$ the structure of an ...
user avatar
7 votes
0 answers
130 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 ...
user avatar
1 vote
0 answers
76 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 ...
user avatar
2 votes
0 answers
68 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 ...
user avatar
0 votes
1 answer
122 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 ...
user avatar
1 vote
0 answers
90 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 ...
user avatar
1 vote
0 answers
101 views

bound norm of algebraic integers in cyclotomic field

Let $\zeta$ be the $p$th root of unity, with $p$ an odd prime number. Let $\mathbb{Q}(\zeta)$ be the $p$th cyclotomic field and let $\mathcal{O}=\mathbb{Z}(\zeta)$ the ring of integers of $\mathbb{Q}(\...
user avatar
  • 19
1 vote
1 answer
214 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 $...
user avatar
2 votes
1 answer
143 views

How can I prove this claim about splitting of prime ideals in real cyclotomic fields?

Let $L_k = \mathbb{Q}(\zeta_{2^k} + \zeta_{2^k}^{-1})$ be the maximal real subfield of the cyclotomic field of conductor $2^k, k \ge 2$ and $f_k(x)$ be the minimal polynomial of $\zeta_{2^k} + \zeta_{...
user avatar
  • 287
6 votes
0 answers
360 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: ...
user avatar
  • 339
2 votes
1 answer
150 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 ...
user avatar
  • 1,190
22 votes
1 answer
756 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 ...
user avatar
10 votes
3 answers
610 views

Realizability of a real representation using real cyclotomic coefficients

Let $G$ be a finite group and $\rho: G \rightarrow GL(d,\mathbb{C})$ an irreducible representation with Frobenius-Schur indicator $\frac{1}{|G|}\sum_{g\in G} \operatorname{tr} \rho(g^2) = 1$. Thus $\...
user avatar
4 votes
1 answer
178 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 ...
user avatar
2 votes
0 answers
115 views

What are all the possible indices for the finite depth subfactors?

Jones index theorem (1983) states that the set of all possible (finite) indices of subfactors is exactly $$\mathrm{Ind}=\{ 4 \cos(\pi/n)^2 \ | \ n \ge 3 \} \cup [4, \infty),$$ but if we restrict to ...
user avatar
9 votes
1 answer
213 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 $n$...
user avatar
  • 8,174
8 votes
0 answers
306 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 $...
user avatar
0 votes
0 answers
122 views

Elliptic units as Euler systems

I’m trying to understand elliptic units in order to work with the Euler systems of the abelian extensions of quadratic imaginary number fields. I’ve looked at few references about the topic, but they ...
user avatar
  • 99
1 vote
0 answers
142 views

Vandermonde shift

I'm looking for any known results on a shift operator commutated by a Vandermonde matrix. That is, let $$T=\begin{bmatrix}0 & 1 & 0 & 0 & \cdots \\ 0 & 0 & 1 & 0 & \...
user avatar
  • 364
1 vote
0 answers
104 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 ...
user avatar
  • 99
2 votes
0 answers
169 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 ...
user avatar
  • 99
5 votes
0 answers
108 views

What is $H^1(\mathbb Z, GL_k(R_n))$ for a ring closely related to the cyclotomic rings of integers?

Let us consider $R_n = \mathbb Z_\ell[\theta_n]/(\theta^{\ell^n}-1)$, an auxiliary prime power $q\equiv 1 \pmod \ell$ with an action of $\mathbb Z = \langle \sigma\rangle$ by $\sigma(\theta_n) = \...
user avatar
  • 6,934
4 votes
0 answers
349 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 ...
user avatar
  • 99
2 votes
0 answers
148 views

Extension of a theorem of Bisch to cyclotomic integers of fixed degree

Theorem 3.2 in this paper (Bisch, 1994) states that for a finite depth ${\rm II}_1$ subfactor of integral index, every intermediate subfactor are also of integral index. As an application, every such ...
user avatar
3 votes
0 answers
120 views

Cyclic codes: sparse codewords not orthogonal to the all-ones vector

Is it true that for any sufficiently large prime $p$, there exists a prime $q\ne p$ and a cyclic code of length $p$ over $\mathbb{F}_q$ that contains a codeword of Hamming weight at most ord$_p(q)$ ...
user avatar
  • 93
6 votes
0 answers
114 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{...
user avatar
  • 161
10 votes
0 answers
476 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&...
user avatar
4 votes
1 answer
211 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 ...
user avatar
3 votes
0 answers
90 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 ...
user avatar
1 vote
0 answers
126 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+...
user avatar
  • 8,174
1 vote
1 answer
161 views

Factoring cyclotomic polynomials over quadratic subfield

The quadratic subfield of $\mathbb{Q}(\zeta_p)$ is given by $\mathbb{Q}(\sqrt{p^*})$, where $p^*$ is the choice of $\pm p$ which is $1$ mod $4$. By some elementary Galois theory, the cyclotomic ...
user avatar
  • 6,049
2 votes
1 answer
144 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{...
user avatar
3 votes
0 answers
115 views

Extended cyclotomic criterion for unitary categorification

According to this paper (Corollary 8.54) the Frobenius-Perron dimension (FPdim) of any object $a$ of a fusion category over $\mathbb{C}$ is a cyclotomic integer. Now, FPdim($a$) is the maximal ...
user avatar
3 votes
0 answers
135 views

Real root of the derivative of a prime cyclotomic polynomial

Consider the graph of $y=x^n$ with $n$ odd, and draw a tangent to its negative arc that crosses the graph at $(1,1)$. Equating the slopes gives us $\frac{x^n-1}{x-1}=nx^{n-1}$ at the point of tangency....
user avatar
  • 1,529
9 votes
1 answer
304 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 ...
user avatar
  • 3,120
1 vote
1 answer
222 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 ...
user avatar
10 votes
1 answer
410 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\...
user avatar
27 votes
2 answers
1k 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 $\...
user avatar
  • 2,133
13 votes
1 answer
287 views

Products of Cyclotomic Polynomials with Nonnegative Coefficients

I'm curious if there are any results that allow us to determine if a product of cyclotomic polynomials (not necessarily all distinct) results in a polynomial having nonnegative coefficients. Some ...
user avatar
10 votes
0 answers
291 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(\...
user avatar
3 votes
1 answer
198 views

About real abelian number fields

How can I prove this: Let $K$ be a real abelian number field, $K_1$ be the Hilbert Class Field of $K$, and $J=K_1\cap K(\zeta_b)$. If a prime $p$ divided $[J:K]$ but did not divide $[K:\mathbb{Q}]$, ...
user avatar
6 votes
1 answer
681 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 ...
user avatar
  • 183
19 votes
3 answers
907 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 ...
user avatar
6 votes
0 answers
67 views

Deciding positivity of real cyclotomic numbers efficiently

Consider a cyclotomic field $\mathbb{Q}[\zeta_n]$ for fixed $n$ and assume that an embedding $\mathbb{Q}[\zeta_n] \hookrightarrow \mathbb{C}$ has been chosen, say by fixing once and for all $\zeta_n=\...
user avatar
2 votes
0 answers
198 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 ...
user avatar
6 votes
2 answers
265 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 $H/H^+$ ...
user avatar
  • 6,934
9 votes
0 answers
413 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 ...
user avatar
  • 6,934
6 votes
1 answer
192 views

Regulator of abelian extensions of Q

Let $K = \mathbb Q(\mu_m)$ and $\zeta_K$ it's Dedekind zeta function. We know from the class number formula that, around $0$: $$\zeta_K(s) \sim s^{r_1+r_2-1}h(K)R(K)/w(K) $$ where $h,R,w$ stand for ...
user avatar
  • 6,934
1 vote
0 answers
188 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 $$f_{d,n}(x)=\frac{x^n-1}{\Phi_d(x)}=a_0+a_1x+\cdots+a_{\deg(f_{d,n}(x))}x^{\deg(f_{d,n}(x))}.$$ ...
user avatar
  • 13k