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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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Products of different cyclotomic polynomials

Is there a classification of all products of different cyclotomic polynomials with non-negative coefficients? Clearly if the cyclotomic polynomials have only nonnegative coefficients, their products ...
11
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1answer
170 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 ...
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0answers
217 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
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1answer
181 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 $[K:\mathbb{Q}]$, ...
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1answer
206 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 ...
19
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3answers
686 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
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0answers
59 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=\...
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0answers
103 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 ...
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2answers
164 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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0answers
162 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
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1answer
160 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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180 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 $$f_{d,n}(x)=\frac{x^n-1}{\Phi_d(x)}=a_0+a_1x+\cdots+a_{\deg(f_{d,n}(x))}x^{\deg(f_{d,n}(x))}.$$ ...
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1answer
175 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
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2answers
176 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
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2answers
380 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
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0answers
249 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
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3answers
512 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
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0answers
150 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
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1answer
325 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 ...
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0answers
38 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 ...
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1answer
310 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? ...
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1answer
111 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
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1answer
385 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
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1answer
786 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 ...
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0answers
119 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
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1answer
617 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)=...
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0answers
513 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
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1answer
205 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
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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
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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 $...
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1answer
292 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 ...
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2answers
308 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 ...
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1answer
371 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
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2answers
552 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
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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
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3answers
1k 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 ...
24
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3answers
7k views

Quick proof of the fact that the ring of integers of Q(\zeta_n) is 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
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2answers
714 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 <...