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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questions with no upvoted or accepted answers
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Evaluating products of cyclotomic polynomials at roots of unity
Are there general non-trivial conditions on $p(\cdot)$ and $n$, where $p(\cdot)$ is a product of cyclotomic polynomials and $n$ is a positive integer, such that all the coefficients of $p(\cdot)$ are ...
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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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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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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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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 ...
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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 ...
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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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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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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 ...
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Meaning of a result of Gauss on "Mensura" of cyclotomic numbers
(This question was asked before on mathstackexchange. I received a few useful comments there, which helped me answer it for a special case, but I did not succeed in proving the general case.)
In an ...
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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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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 $\...
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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 $\...
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Units in Abelian extensions which are not in the subgroup of cyclotomic units
This question is motivated by a Quora post and the top answer to it.
The post wishes to make explicit the difference between units in cyclotomic fields and cyclotomic units.
One problem with answering ...
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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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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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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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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 $...
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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 $\...
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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 $<\...
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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 ...
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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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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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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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Prime factors of $\sum_{i\in I} \zeta_p^i$
For a rational prime $p>3$, denote by $\zeta$ a fixed primitive root of unity of degree $p$, and let ${\mathbb K}={\mathbb Q}(\zeta)$ be the $p$th cyclotomic field. Consider the set of all non-zero ...
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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 ...
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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 ...
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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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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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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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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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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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Abelian extensions of the rationals
Let $E$ and $F$ be finite abelian extensions of $\mathbb{Q}$ such that $E\cap F=\mathbb{Q}$. (You could take $E$ and $F$ as cyclotomic fields if that makes my question easier.) Set $K:=EF$ (the field ...
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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 ...
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"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 ...
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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}(\...
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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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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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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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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(\...
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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 ...
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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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