Question

Multiplicative order of root of unity modulo prime ideal of cyclotomic field

score 129 · Answer 1 · 2020-08-03 19:03:27

129

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Multiplicative order of root of unity modulo prime ideal of cyclotomic field

Published on 03 Aug 2020 - 19:03

score 24 · Answer 2 · 2020-08-04 16:49:23

24

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Let $a,b \in \Bbb C$, show that $\sum_{k=0}^{n-1} |a+w^kb|=\sum_{k=0}^{n-1} |b+w^ka|$ with $w=\exp\bigg(\dfrac{2i\pi}{n}\bigg)$

Published on 04 Aug 2020 - 16:49

score 193 · Answer 3 · 2020-08-05 08:26:36

193

Views

Product of sines when the angles form a sequence

Published on 05 Aug 2020 - 8:26

score 167 · Answer 4 · 2020-08-06 18:31:57

167

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Proving a result related to $100^{th}$ roots of unity.

Published on 06 Aug 2020 - 18:31

score 105 · Answer 5 · 2020-08-13 15:23:21

105

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How to factorize $x^{p^2}-1$?

Published on 13 Aug 2020 - 15:23

score 95 · Answer 6 · 2020-08-14 17:10:29

95

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Find all $n \in \mathbb{N}$ such that $30^n \equiv 20 \pmod {38}$

Published on 14 Aug 2020 - 17:10

score 41 · Answer 7 · 2020-08-15 23:46:07

41

Views

Complete solutions to $\mathrm{e}^{\mathrm{i} \alpha_1} + \cdots + \mathrm{e}^{\mathrm{i} \alpha_n} = 0$.

Published on 15 Aug 2020 - 23:46

score 108 · Answer 8 · 2026-03-28 02:42:57

108

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Compute $\cos(20^\circ)\cos(40^\circ)\cos(80^\circ)$ using roots of unity .

Published on 28 Mar 2026 - 2:42

score 114 · Answer 9 · 2020-08-25 17:34:46

114

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Doubt: $\prod_{b=1}^n(1+\zeta^{ab})=2.$ [Putnam 2015 A3]

Published on 25 Aug 2020 - 17:34

score 98 · Answer 10 · 2020-08-26 02:03:23

98

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Doubt :$\prod_{b=1}^n(1+\zeta^{ab})=2^k$, where $\zeta = e^{2\pi i/n}$ , $n$ is an odd integer, gcd$(a,n)=k$.

Published on 26 Aug 2020 - 2:03

score 698 · Answer 11 · 2026-03-25 16:13:58

698

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Prove that $\sum _{x=0}^{p-1}e^{\frac {2\pi ix^{2}}{p}}={\sqrt {p}} $ , $ p \equiv 1{\pmod {4}}$

Published on 25 Mar 2026 - 16:13

score 211 · Answer 12 · 2020-09-04 17:38:33

211

Views

This finite sum involving roots of unity is bounded, but why?

Published on 04 Sep 2020 - 17:38

score 44 · Answer 13 · 2020-09-04 18:32:58

44

Views

Is it possible to find a closed form for $A(x, p)=\frac{z}{x-z}+\frac{z^{p-1}}{x-z^{p-1}}$?

Published on 04 Sep 2020 - 18:32

score 59 · Answer 14 · 2020-09-04 21:15:23

59

Views

Find a closed form formula for $P(X\in k\mathbb{Z})$, for $X\in\text{Po}(\gamma)$.

Published on 04 Sep 2020 - 21:15

score 194 · Answer 15 · 2020-09-07 11:48:40

194

Views

How to deduce this factorization of $x^5+x+1$ by looking at $\int\frac{3x^4+2x^3-2x+1}{x^5+x+1}dx$?

Published on 07 Sep 2020 - 11:48

List Question

Multiplicative order of root of unity modulo prime ideal of cyclotomic field

Let $a,b \in \Bbb C$, show that $\sum_{k=0}^{n-1} |a+w^kb|=\sum_{k=0}^{n-1} |b+w^ka|$ with $w=\exp\bigg(\dfrac{2i\pi}{n}\bigg)$

Product of sines when the angles form a sequence

Proving a result related to $100^{th}$ roots of unity.

How to factorize $x^{p^2}-1$?

Find all $n \in \mathbb{N}$ such that $30^n \equiv 20 \pmod {38}$

Complete solutions to $\mathrm{e}^{\mathrm{i} \alpha_1} + \cdots + \mathrm{e}^{\mathrm{i} \alpha_n} = 0$.

Compute $\cos(20^\circ)\cos(40^\circ)\cos(80^\circ)$ using roots of unity .

Doubt: $\prod_{b=1}^n(1+\zeta^{ab})=2.$ [Putnam 2015 A3]

Doubt :$\prod_{b=1}^n(1+\zeta^{ab})=2^k$, where $\zeta = e^{2\pi i/n}$ , $n$ is an odd integer, gcd$(a,n)=k$.

Prove that $\sum _{x=0}^{p-1}e^{\frac {2\pi ix^{2}}{p}}={\sqrt {p}} $ , $ p \equiv 1{\pmod {4}}$

This finite sum involving roots of unity is bounded, but why?

Is it possible to find a closed form for $A(x, p)=\frac{z}{x-z}+\frac{z^{p-1}}{x-z^{p-1}}$?

Find a closed form formula for $P(X\in k\mathbb{Z})$, for $X\in\text{Po}(\gamma)$.

How to deduce this factorization of $x^5+x+1$ by looking at $\int\frac{3x^4+2x^3-2x+1}{x^5+x+1}dx$?

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