# Questions tagged [roots-of-unity]

The tag has no usage guidance.

54 questions
Filter by
Sorted by
Tagged with
76 views

247 views

### Möbius inversion formula and roots of unity

Is the exact value of $$\sum_{d\mid n} \mu\left(\frac{n}{d}\right) \zeta^d$$ known? Here, $\mu$ denotes the Möbius function and $\zeta$ a root of unity. At first sight, it seems to me that this ...
246 views

107 views

### Countable roots of unity

I recently learned about non-standard analysis and have the following question. Take the rational numbers; there is a maximal cyclotomic extension (containing all roots of the multiplicative identity)....
430 views

179 views

### Infinitude of cyclotomic polynomials with a certain number of terms

Let $\Phi_n$ be the $n$th cyclotomic polynomial: $${\Phi _{n}(x)=\!\!\prod _{\substack {1\leq k\leq n \\ \gcd(k,n)=1}} \!\!\big(x-e^{2i\pi {k/n}}\big).}$$ Here is a list of the first 30 cyclotomic ...
491 views

### Summation formulas involving roots of unity to various powers

I want to know properties of the following sum: $$\sum_{j=0}^{p-1} \omega^{\beta j^2}= ~?$$ where $p$ is a prime, and $\omega^p=1$, is a $p$th root of unity (and $\beta$ is an integer between $0$ and ...
112 views

196 views

### How to prove an approximation of a combinatorics identity

How to prove that $$\sum_{k\ge 0} \binom{n}{rk} =\frac{1}{r}\sum_{j=0}^{r-1}(1+w^j)^n$$ can be approximated as $\frac{2^n}{r}$, where $n\ge 0$, $r\ge 0$, $n>r$ and $w^r=1$ is a primitive $r$-th ...
227 views

### How small can the nonzero sum of $O(\log n)$ distinct $n-$th roots of unity be?

The OEIS sequence oeis.org/A108380 gives the least number of distinct n-th roots of unity summing to the smallest possible nonzero magnitude. This sequence seems to imply that the least number of ...
593 views

314 views

### Trace of roots of unity has valuation more than 1 in uramified field

Let $F$ be a finite extension of $\mathbb{Q}_p$ (p is prime) and $K/F$ be a unramified extension of prime degree $\ell (\neq p)$. Denote $\mu_K$ be the group of roots of unity in $K.$ Does there exist ...
822 views

### Power sums of p-th roots of unity

The following question was asked by a colleague of mine. For any prime $p$ consider $$M_p:=\min_{z_1,\dots,z_p}\max_{j,k}\left|z_1^k+\dots+z_j^k\right|,$$ where $z_1,\dots,z_p$ are the complex $p$-th ...
545 views

404 views

### q-th powers and roots of polynomials

Let $p,q,r$ be integers with $r\ge2$; let $f$ be a polynomial of the form $f(X) = g((X+1)^r)$, which is not a $q$-th power. Let $\omega$ be a $p$-th root of unity. Prove or disprove that the ...
87 views

1k views

### Probability a polynomial has a root which is a root of unity

Consider a degree $n$ polynomial $P(x)$ with coefficients $c_i \in \{-1,0,1\}$ chosen uniformly and independently. What is the probability that $P(x)$ has a root which is a root of unity? ...
2k views

### An algebraic number is not a root of unity?

This problem is related to my study of the Burau representation of the braid group $B_3$: I was trying to show that certain "congruence subgroups" are of infinite index. There is an approach that ...
774 views

### Minimal polynomial of sums of roots of unity with constant term $\pm1$

Let prime $p$ and given $\zeta_p = e^{2\pi i/p}$. It is well-known that the minimal polynomial of $x = \zeta_p + \zeta_p^{p-1}$ has a constant term either $\pm 1$ and, for certain $p$, the sum of ...
718 views

### Are all sums of subsets of roots of unity unique?

For a prime $p$, let $S$ be the set of all $p$-roots of unity in the complex plane. Now, consider the sum, $W(R)$ of the members of a set $R$ which is a proper subset of $S$. I suspect that $R \ne R'$ ...
Let $m>2$ be an integer and $k=\varphi(m)$ be the number of $m$-th primitive roots of the unity. Let $\Phi = \{ \xi_1, \ldots, \xi_{k/2}\}$ be a set of $k/2$ pairwise distinct primitive $m$-th ...