# 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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### 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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### 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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### 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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### 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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### 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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### 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 ...