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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Quick proof of the fact that the ring of integers of $\mathbb Q(\zeta_n)$ is $\mathbb Z[\zeta_n]$?
I cannot find a good reference for the proof that the ring of integers in a cyclotomic field $\mathbb{Q}(\zeta_n)$ is $\mathbb{Z}[\zeta_n]$. The proof I usually find does an induction on the number of ...
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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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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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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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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 ...
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Can a sum of roots of unity be an integer?
Let $n \geq 2$, $H \lneq (\mathbb{Z}/n\mathbb{Z})^*$, $\zeta_k$ a primitive $k$-th root of unity. Is it possible that $$\sum_{h \in H} \zeta_k^{h} \in \mathbb{Z}$$ for every $k$ dividing $n$ such that ...
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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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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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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
$$ (a_0+a_1\zeta+\dotsb+a_{10}\zeta^{10})(b_0+...
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Realizability of a real representation using real cyclotomic coefficients
Let $G$ be a finite group and $\rho: G \rightarrow GL(d,\mathbb{C})$ an irreducible representation with Frobenius-Schur indicator $\frac{1}{|G|}\sum_{g\in G} \operatorname{tr} \rho(g^2) = 1$. Thus $\...
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Is there a pattern for the irreducible factors and degrees of a characteristic polynomial?
Let $Log(n) = \sum_{i=1}^r \alpha_i \cdot e_i$, where $n = \prod_{i=1}^r p_i^{\alpha_i}$ and $p_i$ is the $i$-th prime, $\alpha_i \ge 0$, $e_i$ is the $i$-th standard basis vector. For example $6 = 2\...
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Tables of class numbers of cyclotomic fields
Does anyone have a table of the class numbers ($h_n$) of cyclotomic fields (upto say, n = 250-300 for $\mathbb Q(\mu_n)$)?
I can find tables for the relative class number ($h_n^-$) in various places ...
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Is class group of cyclotomic fields cyclic?
What are the cyclotomics fields with a cyclic class group. I read that there are only 29 cyclotomic extensions of $\Bbb Q$ with class number one. But I wanted to know what condition on $n$ would make $...
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Toward a cyclotomic Riemann hypothesis
For an integer $n \ge 3$, consider the function $$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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which algebraic integers in a cyclotomic field give you integer absolute value?
Does anyone know an answer to this question?
Question: In an cyclotomic field which algebraic integers have integer absolute value?
Revision 1: -1
I like to add this to the above question, Let's ...
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How can I prove that a sequence of squares of graph norms is never cyclotomic?
The norm of a graph is the largest eigenvalue of the adjacency matrix. I'll write ||G|| for the norm of G.
Now, fix some graph <...
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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 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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Ring of algebraic integers in a quadratic extension of a cyclotomic field
Hello,
I have a question which arose when trying to classify orders of certain algebras.
We know that if $K=\mathbb{Q}(\zeta)$ is any cyclotomic field, and $\zeta$ is an $n$-th root of unity (for ...
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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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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,
$$\frac{z^5}{\beta} = 10 z^3 - 20 n^2 z^2 + 5 (3 n^4 - 25 n^...
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Elements of absolute value 1 in cyclotomic extension of dyadic rationals
Let $\zeta$ be a primitive $2^k$th root of unity, and consider the ring $ \mathbb{Z}[\zeta, \frac{1}{2}] \subset \mathbb{C}$. Are there any elements of absolute value 1 other than the powers of $\zeta$...
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Finding when a certain product in a cyclotomic field is equal to one
For the following, fix $m\geq 2$, and let $a_{0},\dots,a_{m-1}\in\mathbb{Z}$ be such that $\sum_{k=0}^{m-1}a_{k}=0$. I would like to find the exact conditions on the $a_{k}$ so that the following ...
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Trace of n-th root of unity in cyclotomic extension of p-adic rationals
Let $n\in\mathbb N$ and $p$ be any prime. Denote by $\mathbb Q_p$ the $p$-adic numbers. For a field extension $L/K$ denote by $Tr_{L/K}$ the corresponding trace function.
Let $\zeta_n$ be a primitve $...
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Which criteria for "uniformly splitting" polynomials?
Let $P(x)$ be an irreducible monic polynomial of degree $\ge4$ with integer coefficients. We all know that over a finite field $\mathbb F_p$, $P$ will often split, and I am interested in polynomials ...
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Regulator of abelian extensions of Q
Let $K = \mathbb Q(\mu_m)$ and $\zeta_K$ it's Dedekind zeta function. We know from the class number formula that, around $0$:
$$\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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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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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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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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Products of cyclotomic polynomials
Is $\Phi_5(z) \Phi_6(z) = 1 + z^2 + z^3 + z^4 + z^6$ the only product of cyclotomic polynomials that has nonnegative coefficients and satisfies $p(\zeta)=0$, $p(\zeta^2)=2$, $p(\zeta^3)=3$, and $p(1)=...
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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 ...
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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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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 ...
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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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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 ...
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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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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 ...
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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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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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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 ...