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Tagged with primitive-roots complex-numbers
17
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Find sum of primitive roots of $z^{36} − 1 = 0$ [closed]
I am trying to understand this concept of sum of primitive roots of unity and here is a typical problem based on it.
$z^{36} − 1 = 0$
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Powers of primitive roots.
Let $w= \cos\frac{2k\pi}{n} + i \sin\frac{2k\pi}{n}$ be a primitive $n^{\text{th}}$ root of unity, ie, $w^n=1$ and $w^m \neq 1$ for $m \leq n$. Then the powers
$$1, w, w^2, \ldots, w^{n-1}$$
are all ...
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Sum of complex roots' fractions
According to this:
If $\omega^7 =1$ and $\omega \neq 1$ then find value of
$\displaystyle\frac{1}{(\omega+1)^2} +
\frac{1}{(\omega^2+1)^2} +
\frac{1}{(\omega^3+1)^2} +
... + \frac{1}{(\omega^6+1)^2}=...
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3
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what is the value of $\binom{n}{1}+\binom{n}{4}+\binom{n}{7}+\binom{n}{10}+\binom{n}{13}+\dots$
what is the value of $$\binom{n}{1}+\binom{n}{4}+\binom{n}{7}+\binom{n}{10}+\binom{n}{13}+\dots$$ in the form of number,
cos, sin
attempts : I can calculate the value of $$\binom{n}{0}+\binom{n}{...
1
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1
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Primitive roots of the unity in $\mathbb C$
Let $\omega$ be a primitive $n-$th root of unity.
(i) Show that its powers $\omega^k$, for $k ∈ {1, \ldots, n}$, are all different;
(ii) Deduce that they are precisely all the $n-$th roots of unity.
...
user720013
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Let $w \in G_{63}$ be a primitive root of unity. Find all $n\geq 6$ such that $\sum_{k=6}^n w^{35k} = 0$ and $w^{12n} = w^{15}$
This is probably totally wrong.
We know that $5$ and $63$ are relatively prime, therefore $w^5 \in G_{63}$ primitive (we'll suppose $w= w^5$ without loss of generality).
We also know that $(w^7)^9 = ...
- 781
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Let $w\in G_{18}$ be a primitive root of unity. Prove that $w^{16} \sum_{j=1}^{17}(w^3 \overline{w})^{3j+1}$ is imaginary pure.
This is what I've got:
$$w^3\overline{w}=w^3w^{-1}=w^2 \iff \\ w^{16} \sum_{j=1}^{17}(w^3 \overline{w})^{3j+1} = w^{16}w^2\sum_{j=1}^{17}w^{6j} = \bigg( \sum_{j=0}^{17}w^{6j} \bigg) - 1$$
Given that ...
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Let $w \in G_{15}$ be a primitive root. Find every $n \in \mathbb{N}$ such that $\sum_{i=2}^{n-1} w^{3i} = 0$
We can first rewrite the series in a useful form,
$$\sum_{i=2}^{n-1} w^{3i} = \bigg( \sum_{i=0}^{n-1} w^{3i} \bigg) - w^3 - 1 $$
But since $w$ is primitive, we can apply the geometric series formula,...
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Prove that $\eta - \omega \notin \mathbb{Q}$ where $\omega$ and $\eta$ are two differents n-th primitive roots $\in \mathbb{C}$
Let $n \in \mathbb{N}$ be a natural number, and be $\omega$ and $\eta$ two differents n-th primitive roots in $\mathbb{C}$.
Prove that $\eta - \omega \notin \mathbb{Q}$
My attempt was to follow the ...
- 5,164
1
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0
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Primitive nth roots of unity related to the complex nth roots of 1.
I just cannot seem to wrap my head around this problem and would really appreciate some guidance. So I know that a primitive $n^{th}$ root of unity is a complex number $z$ such that $z^n = 1$ but $z^m ...
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Let $w$ be a primitive root of a unit of order 3, prove that $(1-w+w^2)(1+w-w^2)=4$
The title is the statement of the problem.
I did the following:
$(1-w+w^2)(1+w-w^2)=$
$1+w-w^2-w-w^2+w^3+w^2+w^3-w^4=$
$1-w^2+w^3+w^3-w^4=$
$1-w^2+1+1-w^4=4$, * then,by definition of primitive ...
- 698
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Modulo arithmetic and sum of arbitrary powers of a primitive root of unity.
Prove that if $w$ is a primitive nth root of unity, then
$1 + w^k + (w^k)^2 + (w^k)^3 + \cdots + (w^k)^{n-1} =0$ iff $k \neq 0$ mod $n$.
Sorry for the terrible formatting. Also, I don't know ...
- 149
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Express complex number in terms of radicals
Let $\zeta=\cos(\frac{2\pi}{16})+i\sin(\frac{2\pi}{16})$ be a 16th root of unity, so that it is a primitive root of unity. I need to explicitly express this number in terms of radicals: $a+ib$, where $...
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Find $\sum_{i=0}^{k-1}w^{in}$ in terms of $n \in \Bbb N$ being $w \in \Bbb C$ a k-th primitive root of unity.
I need some help with the following problem:
Find $\sum_{i=0}^{k-1}w^{in}$ in terms of $n \in \Bbb N$ being $w \in \Bbb C$ a k-th primitive root of unity.
I thought of writing it as: $\frac{w^{nk}-...
- 1,518
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Help with problem: Let $w$ a 15th-primitive-root of unity. Find all $n \in \Bbb N_{<0}$ such that $\sum_{i=0}^{n-1} w^{5i}=0$
we are starting to see complex numbers in my algebra class. So I have the following problem:
Let $w$ a 15th-primitive-root of unity. Find all $n \in \Bbb N_{<0}$ such that $\sum_{i=0}^{n-1} w^{...
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