# 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

71 views

### Why is $\mathbb F_5(\root{15}\of t)$ not normal over $\mathbb F_5(t)$?

Why is $\mathbb F_5(\root{15}\of t)$ not normal over $\mathbb F_5(t)$? (I'm asking this question in order to understand this answer). My idea for a proof so far: $f:=X^{15}-t\in F_5(t)$ is the ...
34 views

### Considerations on cyclotomic extensions

I stopped in front of some issues regarding certain passages of the theorem 9.4 of Algebra of Serge Lang, in which we suppose to have k a field such that $[k (\mu_n):k]=\phi (n)$ where $\mu_n$ is ...
114 views

### Find the minimum polynomial of a sum of roots of unity.

Let $\omega$ be an 11-th primitive root of 1 over $\Bbb Q$ Let $\beta = \omega + \omega^9$ Find $[ \Bbb Q ( \beta) : \Bbb Q ) ]$ and Find the minimum polynomail of $\beta$. I asked a ...
83 views

### Radical expression for roots of unity

Can somebody point out a reference to the nested radical formula of the complex roots of unity when $n = 2^N$, i.e. in solving $x^n=1$ ?
67 views

31 views

### argument of a lacunary sum of roots of unity

Let $q>4$ and $t< \sqrt{q}$ be integers. Determine the set $\{j_1,...,j_t\}$ of integers $0 \leq j_i <q-1$ such that $\arg(\sum_{i=0}^t e^{2i\pi\frac{j_i}{q}} ) \in [0,\pi[$.
191 views

### sum of $p$th roots of unity

Let $p$ be an odd prime. Let $\mathbb Z_p$ be the ring of integers from $0$ to $p-1$, i.e. $\mathbb Z_p=\left\{0,1,\dots,p-1\right\}$. Let $r$ be a positive integer. Then, for all values of $p$, is ...
133 views

19 views

### Norm of roots of unity in a subextension.

Suppose $L/k$ is a Galois extension of number fields, $G$ is the Galois group, and for $\frak p$ a prime ideal of $\cal O$$_L$, let $K=L^{G(\frak p)}$, where [$L$ : $K$] = $p$. Suppose $m$ is the ...
31 views

### Complex roots for integer polynoms $p(z)=0$ with restriction $| z|=1$

There have been may questions about (integer) polynomials of sin and cos. There have been nearly as many ingenious answers using trigonomic-magic (converting sin, cos, tan into each other), but I (not ...
29 views

### $(ab^{n+1})^{1/n}$ where $a,b \in \mathbb{C}$

Let $a,b \in \mathbb{C}$ and $n \in \mathbb{N}$. We can present $a$ and $b$ in polar form as a = r_a \mathrm{e}^{i \theta_a} \quad \textrm{and} \quad b = r_b \mathrm{e}^{i \theta_b}, ...
98 views

### Carmichael function and primitive roots of unity

I have been reading about the Carmichael function recently and I would like to ask about some elementary implication of its properties as I haven't found it stated explicitly. If I understand it ...
Let $A$ be a DVR such that its fraction field $K$ is complete w.r.t to the natural absolute value in $K$. I am trying to prove that the projection from $A$ to the residue class field, $F$, maps the ...
### Isomorphism betwixt a Galois Group and the set of $n$th roots of unity
Let $p \in \mathbb{P}$ and $n \in \mathbb{N}: p \nmid n$; the set of $n$th roots of unity be $W_n$; $\mathbb{F}$ be a field$: char(\mathbb{F}) \in \left\{ {0,p}\right\}, \mathbb{F} \supseteq W_n$ (set ...