0
votes
0answers
21 views

$b_k = \sum\limits_{j=0}^4 j\omega^{-kj}$, for $0\le k\le4$ $\Rightarrow$ $\sum\limits_{k=0}^4 b_k\omega^k$ =?

Let $\omega$ denote a complex fifth root of unity. Define $b_k = \sum\limits_{j=0}^4 j\omega^{-kj}$, for $0\le k\le4$. Then find the value of $\sum\limits_{k=0}^4 b_k\omega^k$.
1
vote
1answer
59 views

Galois Group of $x^n - a$

Homework problem: If the field F contains a primitive nth root of unity, prove that the Galois group of $x^n - a$, for $a \in F$, is abelian. I'm not really sure where to start here and I'm ...
0
votes
1answer
159 views

Proof with roots of unity

Let $m,n \in \mathbb N$ and $d=gcd(m,n).$ Prove that if w is both an m-th root of unity and an n-th root of unity, then w is a d-th root of unity. How would i begin about starting this type of proof? ...
0
votes
1answer
35 views

Roots of unity product

For each $n \in \mathbb N, n \geq 3$ calculate the product of all the n roots of unity. Or to say it in a more stric way: $$\prod_{w \in G_n^*}w$$ Being $G_n^*$ the primitive roots of the unity.
0
votes
2answers
289 views

Proving equations involving the powers of a complex cube root of unity ω

The question in this homework problem is to show $ω^4 + ω^5 = -ω^6$ given that $ω$ is a complex cube root of unity. I am also required to show that $(1 - ω)^2 = -3ω$, but if I am assisted with ...
0
votes
1answer
51 views

Adjoining two primitive n-th roots

Let $\omega_n$ denote a primitive $n^{th}$ root of unity. If $m$ and $n$ are positive integers with $lcm(m,n)=k$, show that $\mathbb{Q}(\omega_n,\omega_m)=\mathbb{Q}(\omega_k)$. To start, I am aware ...
4
votes
1answer
314 views

A finite field extension $K\supset\mathbb Q$ contains finitely many roots of unity

I must show that any finite field extension $K\supset \mathbb Q$ can only contain finitely many roots of unity. I reasoned in the following manner: Let $n<\infty$ be the degree of $K$ over ...
1
vote
3answers
111 views

Subtracting roots of unity. Specifically $\omega^3 - \omega^2$

This is question that came up in one of the past papers I have been doing for my exams. Its says that if $\omega=\cos(\pi/5)+i\sin(\pi/5)$. What is $\omega^3-\omega^2$. I can find $\omega^3$ and ...
5
votes
2answers
306 views

Radical extension

Let $K=\mathbb{Q}(\sqrt[n]a)$ where $a\in\mathbb{Q}$, $a>0$ and suppose $[K:\mathbb{Q}]=n$. Let $E$ be any subfield of $K$ and let $[E:\mathbb{Q}]=d.$ Prove that $E=\mathbb{Q}(\sqrt[d]a)$. It's ...
0
votes
1answer
169 views

Roots of unity in a field generated by a root of a polynomial

The polynomial x^3 − 2 has no roots in F7 and is therefore irreducible in F7[x]. Adjoin a root k to make the field F := F7(k), which will be of degree 3 over F7 and therefore of size 343. The ...
1
vote
2answers
293 views

show the rule between $w$, $w^2$, …$w^{n-1}$ with $w^n$ given w an nth root of unity

Let $w≠1$ be an $n$-th root of unity, i.e., $w^n-1=0$. Show that $1+2w+3w^2+\dots+nw^{n-1}=-\frac n{1-w}$. My question is how to relate $w, w^2, \dots,w^{n-1}$ with $w^n$?
0
votes
1answer
137 views

Cyclotomic Polynomials

Let $E(n)$ denote an nth root of unity. (For convenience, we may take $E(n) = \exp(\frac{2πi}{n})$.) Prove that for any prime $p$ and any natural number $r$, we have $$ \prod_{\substack{j\\ \gcd(p^r, ...
2
votes
1answer
129 views

Fields modulo $n$-th powers, discrete valuations and roots of unity

I have been doing some revision on local field theory and have gathered up a collection of questions which I have been unable to make much progress with; there will be a few similar queries along with ...