# Questions tagged [roots-of-unity]

numbers $z$ such that $z^n=1$ for some natural number $n$; here usually $z$ is in $\mathbb C$ or some other field

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### Proof of $\frac{\Phi_n'(x)}{\Phi_n(x)} = \frac{1}{x^n-1}\sum_{k=1}^{n}c_k(n)x^{k-1}$

Let $c_k(n)$ denote Ramanujan's sum, and $\Phi_n(x)$ be the $n$th cyclotomic polynomial. Prove that $$\frac{\Phi_n'(x)}{\Phi_n(x)} = \frac{1}{x^n-1}\sum_{k=1}^{n}c_k(n)x^{k-1}$$ My attempt was to ...
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### $\overline{\mathbb{F}_2}$ does not contain a primitive 10th root of unity

I need to prove/disprove the following statement: Every algebraically closed field $K$ contains a 10th root of unity. I don't think the statement is true. My counterexample is as follows: Let's take ...
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### Natural map of automorphism groups

Question: Write $\mathbb{Q}(\zeta_{\infty}) = \mathbb{Q}(E)$, where $E$ is the group of roots of unity in $\mathbb{Q}^{*}$. Prove that $\mathbb{Q} \subset \mathbb{Q}(\zeta_{\infty})$ is Galois, and ...
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### Resources on infinite series involving $n$'th roots of unity

Background I was wondering whether there any books, articles, or other in-depth treatments of infinite series involving the roots of unity. Let $\omega_{n} := e^{2 \pi i / n}$ be the $n$'th root of ...
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### Understanding roots of unity in quadratic fields

Suppose we have a quadratic field $\mathbb{Q}(z)$ with $z \in \mathbb{C} \setminus \mathbb{Z}$ and $z^2 \in \mathbb{Z}$. How would one go about determining the possible $n^{\text{th}}$roots of unity ...
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### Linear Dependence of Primitive Roots of Unity

Consider the cyclotomic field $\mathbb{Q}(\zeta_n)$. We know that the set of primitive roots $\Pi_n=\{\zeta_n^m:(m,n)=1\}$ generates $\mathbb{Q}(\zeta_n)$ as a field. However, what happens when we ...
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### Permutating the coeffecients of equations with roots of unity.

I"ve noticed the roots of $$(x^k-1 =0) = (1-x^k=0)$$ This is simple enough since changing +/- signs gives you the additive inverse which is equal at zero. However this property can be also be ...
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### Calculating Coefficents of a single variable polynomial [duplicate]

Given: $$(1+x+x^2+x^3+\cdots+x^k)^n$$ Is there a formula to calculate the coefficient of $x^a$ (where $a$ can be any integer value less than $k^n$) that's more efficient than grinding through ...
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### Algebraic forms for nth Roots of unity

Looking for algebraic forms for primitive nth roots of unity when n is composite. When n is prime it is simply all the possible roots of $\sum_{m=0}^{\phi(n)} X^M =0$ It's the composite values ...
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### A Dilogarithmic Sum

I have been playing around with dilogarithm and considered the following sum. $$\sum_{n=1}^{k} {\text{Li}_{2} {(\omega^n)}}$$ Where $\text{Li}_2{(x)}$ is the Dilogarithm and $\omega$ is the k-th root ...
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### $2$ as order $n$ unitroot in $\mathrm{GF}(2^n \pm 1)$: what should I know?
I am studying a fake proof of FLT which assumes $\mathbb Q\left[\zeta_p \right]$ is a UFD (that is, that the rationals with the pth root of unity is a UFD). We have covered the proofs for n=2, 3, and ...