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
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An insight about Euler systems and cyclotomic units
I am trying to understand Euler systems, and their relation to Main Conjectures. Usually it is told to start with cyclotomic Euler systems, since they does not need cohomology techniques. I followed a ...
1
vote
0answers
75 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+...
1
vote
1answer
110 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 ...
2
votes
1answer
118 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{...
2
votes
0answers
91 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 ...
2
votes
0answers
92 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....
0
votes
0answers
88 views
Integers on Pentagrid
in de Bruijn's Pentagrid, the generating lines for Penrose Tiling are done by an intersection of a $2D$ hyperplane with the unit cube in $\mathbb{R}^5$, which formula can be found here:
http://www....
9
votes
1answer
227 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 ...
1
vote
1answer
121 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 ...
10
votes
1answer
377 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\...
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votes
2answers
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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 $\...
12
votes
1answer
210 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 ...
10
votes
0answers
259 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(\...
3
votes
1answer
188 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 $[K:\mathbb{Q}]$, ...
5
votes
1answer
361 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 ...
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votes
3answers
786 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 ...
6
votes
0answers
63 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=\...
2
votes
0answers
166 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 ...
6
votes
2answers
215 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 $H/H^+$ ...
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votes
0answers
250 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 ...
6
votes
1answer
166 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 ...
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185 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 $$f_{d,n}(x)=\frac{x^n-1}{\Phi_d(x)}=a_0+a_1x+\cdots+a_{\deg(f_{d,n}(x))}x^{\deg(f_{d,n}(x))}.$$
...
1
vote
1answer
182 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 ...
7
votes
2answers
199 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 $\zeta$...
10
votes
2answers
420 views
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
$$ (a_0+a_1\zeta+\dotsb+a_{10}\zeta^{10})(b_0+...
3
votes
0answers
313 views
Cyclotomic Extension of a Perfectoid Space
Maybe, I am being stupid, but when I consider ramified extension of a perfectoid field with the characteristic $0$, I cannot find the correspondent field with characteristic $p$. Let me put it more ...
6
votes
3answers
542 views
Something interesting about the quintic $x^5 + x^4 - 4 x^3 - 3 x^2 + 3 x + 1=0$ and its cousins
(Update):
Courtesy of Myerson's and Elkies' answers, we find a second simple cyclic quintic for $\cos\frac{\pi}{p}$ with $p=10m+1$ as,
$$F(z)=z^5 - 10 p z^3 + 20 n^2 p z^2 - 5 p (3 n^4 - 25 n^2 - 625) ...
4
votes
0answers
151 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
1answer
385 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 ...
2
votes
0answers
39 views
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 ...
4
votes
1answer
342 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?
...
2
votes
1answer
120 views
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 ...
5
votes
1answer
387 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
1answer
811 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 ...
3
votes
0answers
124 views
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 ...
5
votes
1answer
712 views
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)=...
11
votes
0answers
548 views
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 ...
6
votes
1answer
223 views
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 ...
13
votes
4answers
3k views
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 ...
5
votes
2answers
1k views
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 $...
3
votes
1answer
307 views
Cyclotomic integers with given modulus
The following problem was posted to the NMBRTHRY mailing list about a week ago, without eventually getting a satisfactory solution.
Suppose that $p=(n^2+1)/2$ is a prime, with $n\ge 5$ integer. Does ...
1
vote
2answers
326 views
Algebraic numbers abhorrent to cyclotomic fields
Consider an algebraic number $\alpha$, which can be taken to be an
integer. With $\deg\alpha$ a prime number, one can easily arrange that
to be such that all powers $\alpha^n$ to be of the same ...
2
votes
1answer
378 views
is there any bound on the absolute number of algebraic integer in terms of its degree?
If Z is a sum of t distinct roots of unity and |Z| is a rational integer, can someone find a bound on |Z| in terms of k=deg(Q(Z):Q))?
Clearly we need to have distinct roots of unity otherwise this ...
2
votes
2answers
557 views
When does the absolute value of a sum of an integer and an algebraic integer equal an integer?
Let's say Z is a sum of n-th roots of unity and thus an algebraic integer, and D is a rational integer. If |z+D| is an integer, what can we conclude regarding Z? Can we say |Z| is an integer?
Another ...
8
votes
3answers
1k views
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 ...
7
votes
3answers
2k views
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 ...
27
votes
3answers
8k views
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 ...
7
votes
2answers
731 views
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 <...
