# Tagged Questions

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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### Trace of roots of unity has valuation more than 1

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 ...
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### minimal polynomial $\zeta_n$ and $\zeta_n^p$ is the same for any prime $p$ not dividing $n$

I want to prove that for any prime $p$ not dividing $n$, $\zeta_n$ and $\zeta_n^p$ have the same minimal polynomial over $\mathbb{Q}$. My proposed proof, Suppose $\zeta_n$ is a primitive $n$th root ...
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### Find all polynomials $P(x)$ such that $P(x^2)=P(x)^2$

Find all polynomials $P:\mathbb{C}\rightarrow\mathbb{C}$ such that $$P(x^2)=P(x)^2 .$$ Here is what I tried: First, it is easy to see the constant solutions, namely $P\equiv 0,P\equiv 1$. Let $r$ ...
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### If $\alpha_1,\alpha_2,\ldots,\alpha_n$ be the roots of the equation $x^n+1$

then $(1-\alpha_1)(1-\alpha_2)\ldots(1-\alpha_n)$ equals to ? I think here we need the info of whether $n$ is even or odd else how will we say whether by vieta's formula what is the value of $1+(-1)^n$...
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### $f(z_0)=\sum_{n=0}^{k-1}z_0^n + 1+1+ \dots$ - Dense on the unit circle

Personal question : Let $z_0$ be a $2^k$th root of unity. We obtain for the function $f(z)=\sum_{n \geq 0} z^{2^n}$ (radius of convergence $R=1$) that $f(z_0)=\sum_{n=0}^{k-1}z_0^n + 1+1+ \dots$. Why ...
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### Solve for $z$ in $z^3=8i$

I only have a question about the end results. I answered the question fully but my professor knocked off 1 point for my $w_1^0$ result, but I don't know why. He circled the $i\pi /6$ in my answer but ...
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### Alternating sum of roots of unity $\sum_{k=0}^{n-1}(-1)^k\omega^k$

Consider the roots of unity of $z^n = 1$, say $1, \omega, \ldots, \omega^{n-1}$ where $\omega = e^{i\frac{2\pi}n}$. It is a well known result that $\sum_{k=0}^{n-1}\omega^k = 0$, but what if we want ...
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### Let $\zeta_n$ be the $n^{th}$ root of unity $\zeta_n=e^{2\pi i/n}$. How can I prove that $\zeta_5\notin \mathbb{Q}(\zeta_7)$?

This question is from Artin 15.3.3: Let $\zeta_n$ be the $n^{th}$ root of unity $\zeta_n=e^{2\pi i/n}$. How can I prove that $\zeta_5\notin \mathbb{Q}(\zeta_7)$? I'm quite stuck so any help would be ...
### roots of unity and cyclotomic polynomials over $\mathbb{F}_p$
Given a prime $p$, Let $n = p^d -1$ and let $f$ be an irreducible polynomial dividing the $n$-th cyclotomic polynomial in $\mathbb{F}_p[t]$. Let $\alpha = t + (f)$ in $\mathbb{F}_p[t]/(f)$ where $(f)$ ...
### Find all $z \in {\mathbb C}$ such that $z^{12}=1$ and $1+z+z^2+z^3+z^4+z^5 \in {\mathbb R}$
So, my first thought. If $z^{12} = 1, z \in$ is a twelfth root of unity. Knowing this, I can write $z = e^{i {2k \pi} \over {12}}$, with k $\in \{0,1,2,3,4,5,6,7,8,9,10,11\}$. Then if I just ...