# 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

640 questions
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### Do there exist non-trivial integer coefficients that break linear independence of the roots of unity?

Let $n$ be a positive integer, $k=0,\cdots,n-1$, $\omega_k=e^{\tfrac{2\pi i}{n}k}$ be the roots of unity, $c_k \in \mathcal{Z}$ be integer coefficients, trivial and non-trivial be two subcategories ...
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### To find the sum: $\frac {1}{n!} \sum \binom {n}{2+3r} x^{1+r}$

Sum the series: $$\frac {x}{2!(n-2)!}+\frac {x^2}{5!(n-5)!}+\frac {x^3}{8!(n-8)!}+....+\frac {x^{\frac {n}{3}}}{(n-1)!},$$ $n$ being a multiple of $3$.(Math. Tripos, 1899) My attempt We may ...
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### If $K$ is a subfield of $\mathbb{C}$ then $K(\zeta)/K$ is Galois

A lemma in my lecture notes states that if $K$ is a subfield of $\mathbb{C}$ and $\zeta=\exp(2\pi i/p)$ then $K(\zeta)/K$ is Galois. They proved it by arguing that the minimal polynomial of $\zeta$ ...
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### If $a$ is an $n$-th root of $z$ , $b$ is an $n$-th root of $w$, and $c$ is an $n$-th root of $zw$, then $ab=c$.

How do I prove that for all natural numbers $n$ and complex numbers $a, b, c, z, w$ if $a$ is an $n$-th root of $z$ , $b$ is an $n$-th root of $w$, and $c$ is an $n$-th root of $zw$, then $ab=c$. ...
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### Showing the lemma $\operatorname{ord}_p(1+ζ_p)=0$ if $p>2$

Just to give some background regarding my motivation, I'm trying to prove a lemma to help me solve How do we prove p-order of $g_k$ is $\frac {k} {p-1}$? Let $Z_p$ denote the p-adic integers, and let ...
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### Proving $ord_p(ζ_p-1)=1/(p-1)$

After proving this, I was able to deduce an even more general result that $ord_p(ζ_p-1)=ord_p(ζ_p^2-1)=...=ord_p(ζ_p^{p-1}-1)$. Now, according to Lubin, $ord_p(ζ_p-1)$ should be $1/(p-1)$, but this ...
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### Proof that there are exactly $n$ distinct $n$th roots of unity in fields of characteristic zero?

I think it's true that in a field $F$ of characteristic zero, there are exactly $n$ distinct $n$th roots of unity (in some algebraic closure $\bar{F}$), that is, roots of the polynomial $x^n-1$. I ...
I am using Ian Stewart Galois theory book and it says that for $A =$ primitive $p^2$ root of unity $A$ has min poly of $m(t)= 1+ t^p +.....+t^{p(p-1)}$ and so $p(p-1)$ is a power of two. why is ...
### Complex Endomorphism $f$ with orthogonal Eigenspaces implies that $f$ is unitary
I already posted this question earlier today without much success. After 3 hours and only 10 views I decided to give it another shot :) Of course I'll delete the second post! Let $V \ne 0$ be an ...