# Questions tagged [roots-of-unity]

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### Sum of roots of unity

Today I came across the series $\sum_{k=0}^{n-1}\varepsilon^{2k^2}$, where $\varepsilon$ is some primitive $n^\text{th}$ root of unity. Is there an explicit expression for this sum? I mistakenly ...
1 vote
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### Norm of $2^{i}$-th primitive root

Let $K$ be finite degree extension of $\mathbb{Q}$ such that $-1$ is not a square in $K$. Let $L = \frac{K[x]}{\langle x^2 +1\rangle}$. Thus every element of $L$ is of the form $a + ib$ ...
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### About nilpotent Jordan algebras, matrix representations and formally real algebras

Given an non-commutative associative unital algebra A of characteristic $0$, one can construct a Jordan algebra $A+$ using the same underlying addition vector space. Notice first that an associative ...
1 vote
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### $\sin(\frac{\pi}{p})$ not expressible by positive radicals and $\sin(\frac{\pi}{q_i})$?

We have the following identities: $\sin(\frac{\pi}{1})=0$ $\sin(\frac{\pi}{2})=1$ $\sin(\frac{\pi}{3})=\frac{\sqrt{3}}{2}$ $\sin(\frac{\pi}{4})=\sqrt{\frac{1}{2}}$ Lets start with a definition. Rules ...
295 views

### A similar relationship between the generic cubic and the Lehmer quintic?

I. Comparison It doesn't seem to be well-known that the generic cubic (prominent in this MO post) for $C_3 = A_3$, $$x^3-nx^2+(n-3)x+1 = 0$$ has the nice property that its roots $a,b,c$, if in correct ...
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### Finite sum involving root of unity

I have the following sum: $$\sum_{\sigma=1,\text{odd}}^{\frac{p}{2}-2}\frac{\sin \frac{r \sigma \pi}{p}}{\sin \frac{q \sigma \pi}{p}}$$ where $p\equiv2\pmod4$, $p$ and $q$ are coprime numbers such ...
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### $q$ as a prime power and a root of unity

The number of points on the $(n-1)$-dimensional projective space $P^{n-1}(\mathbb{F}_q)$ over a finite field $\mathbb{F}_q$ is the $q$-integer $$[n]_q := \frac{q^n-1}{q-1}.$$ In analogy, the number of ...
285 views

### Roots of anti-palindromic polynomial if coefficients are odd.

This is in continuation of the question asked in this earlier post here. Given an anti palindromic polynomial of degree $n$ with odd coefficients, does it have roots on the unit circle?
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### Modular root of $-1$

Take two distinct coprime integers $a,b$ and take $q=a^2+b^2$. Consider the equation $(xy^{-1})^2\equiv-1\bmod q$. The solutions are two roots which correspond to $(x,y)=(a,b)$ and $(x,y)=(b,a)$. Look ...
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### power series and roots of unity

Let $p$ be an odd prime and $X$ and $Y$ be subsets of $p^{th}$ roots of unity, $|X|=|Y|=n,X\neq Y.$ Let $f(t)=\sum_{x\in X}x^{t}-\sum_{y\in Y}y^{t}$. If $f(t)=at^k+o(t^k)$ is the power series ...
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### Homomorphism from integral module generated by roots of unity to cyclic group?

Let $S$ be the set of all roots of unity. Consider the $\mathbb{Z}$-module, $\mathbb{Z} S$, as an additive abelian group (that is, $\mathbb{Z} S$ is the subset of complex numbers that can be ...