# 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.

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### 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:
...

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### 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 ...

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### 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 ...

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### 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 $\...

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### 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 ...

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### 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 ...

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### 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 $n$...

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### 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 $...

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### 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 ...

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### 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 & \...

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### 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 ...

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### 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 ...

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### 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) = \...

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### 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 ...

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### 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 ...

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### 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$_p(q)$ ...

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### 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{...

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### 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&...

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### 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 ...

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### 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 ...

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### 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+...

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### 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 ...

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### 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{...

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### 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 ...

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### 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....

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### 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 ...

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### 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 ...

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### 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\...

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### 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 $\...

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### 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 ...

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### 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(\...

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### 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 $[K:\mathbb{Q}]$, ...

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### 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 ...

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### 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 ...

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### 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=\...

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### 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 ...

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### 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 $H/H^+$ ...

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### 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 ...

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### 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 ...

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### 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 $$f_{d,n}(x)=\frac{x^n-1}{\Phi_d(x)}=a_0+a_1x+\cdots+a_{\deg(f_{d,n}(x))}x^{\deg(f_{d,n}(x))}.$$
...

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### 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 ...

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### 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 $\zeta$...

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### Special units in the $11$th cyclotomic field

In connection with this problem:
Do there exist integers $a_0,\dotsc,b_{10}\ge 0$ such that $a_0+\dotsb+a_{10}=36$, $b_0+\dotsb+b_{10}=37$, and
$$ (a_0+a_1\zeta+\dotsb+a_{10}\zeta^{10})(b_0+...

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### 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 ...

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### 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 simple cyclic quintic for $\cos\frac{\pi}{p}$ with $p=10m+1$ as,
$$F(z)=z^5 - 10 p z^3 + 20 n^2 p z^2 - 5 p (3 n^4 - 25 n^2 - 625) ...

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### 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 ...

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### 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 ...

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### Tensor of Relative Bases

Suppose $U,V,W$ are the cyclotomic fields $\mathbb{Q}(\zeta_{m_3})$, $\mathbb{Q}_(\zeta_{m_2})$, and $\mathbb{Q}(\zeta_{m_1})$ respectively, where $m_1 \mid m_2 \mid m_3$ so that $U/V/W$ is a tower of ...

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### 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?
...

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### covering radius of a lattice from cyclotomic extension

Given a finite field extension $L$ of $\mathbb{Q}$ of dimension $n$, there is a natural way to embed it into $\mathbb{R}^n$ such that the image of its ring of integers $\mathcal{O}_L$ is a lattice. If ...