Question

Do the zeroes of this polynomial lie inside, outside, or on the unit circle? $P_n(z)=1^3 z + 2^3 z^2 + 3^3 z^3 + \cdots + n^3 z^n$

score 799 · Answer 1 · 2026-04-17 01:20:20

799

Views

Do the zeroes of this polynomial lie inside, outside, or on the unit circle? $P_n(z)=1^3 z + 2^3 z^2 + 3^3 z^3 + \cdots + n^3 z^n$

Published on 17 Apr 2026 - 1:20

score 1.6k · Answer 2 · 2026-04-11 08:18:44

1.6k

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What is the sum of the squares of the 10th roots of unity?

Published on 11 Apr 2026 - 8:18

score 87 · Answer 3 · 2026-04-17 12:53:19

87

Views

If $y^4 = 5$ and $z^6 = 15$, then $y \notin \mathbb{Q}(z)$

Published on 17 Apr 2026 - 12:53

score 191 · Answer 4 · 2026-05-10 20:03:58

191

Views

Determine a Normal Basis for Galois Extension of $\mathbb{Q}$ with primitive pth root of unit (p prime)

Published on 10 May 2026 - 20:03

score 458 · Answer 5 · 2026-05-10 14:48:51

458

Views

Finding the Normal Basis of Cyclotomic field

Published on 10 May 2026 - 14:48

score 166 · Answer 6 · 2026-04-15 03:58:53

166

Views

Why is the magnitude of the sum of two adjacent nth roots always an 'interesting' number, and what do these numbers have to do with each other?

Published on 15 Apr 2026 - 3:58

score 238 · Answer 7 · 2026-04-14 23:54:45

238

Views

How to get common roots of unity of $ z^{6}=1 $ and $ z^{21}=1 $?

Published on 14 Apr 2026 - 23:54

score 197 · Answer 8 · 2026-05-10 21:00:53

197

Views

$\zeta_n^k$ is primitive if and only if $(k,n) = 1$

Published on 10 May 2026 - 21:00

score 1k · Answer 9 · 2026-04-11 09:37:18

1k

Views

$ \bigcup_{n=1}^{\infty}\{z\in\Bbb C:z^n=1\}=\{z\in\Bbb C:|z|=1\}$?

Published on 11 Apr 2026 - 9:37

score 89 · Answer 10 · 2026-04-16 16:44:12

89

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Why is $\frac{\cos(\alpha + \beta) + \cos(-\alpha)}{\sin(\alpha + \beta) + \sin(-\alpha)}$ independent of $\alpha$?

Published on 16 Apr 2026 - 16:44

score 2.1k · Answer 11 · 2026-04-14 13:08:37

2.1k

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(Degree of) Splitting Field for $f(x) = x^p - 2$ over $\mathbb{Q}$, p prime

Published on 14 Apr 2026 - 13:08

score 109 · Answer 12 · 2026-04-16 14:39:01

109

Views

Prove that $z_1,z_2,z_3$ with equal, non-zero modulus, are vertices of an equilateral triangle if and only if $\sum_{cyc}|z_1-z_2|(z_1+z_2) = 0 $

Published on 16 Apr 2026 - 14:39

score 92 · Answer 13 · 2026-04-10 18:03:35

92

Views

Kummer ring - special monic polynomial with zero at root of unity

Published on 10 Apr 2026 - 18:03

score 1.3k · Answer 14 · 2026-04-14 05:46:22

1.3k

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Why are roots of unity evenly spaced?

Published on 14 Apr 2026 - 5:46

score 53 · Answer 15 · 2026-05-10 22:04:35

53

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Let $n>2$ prove that $[\mathbb Q(z+\frac{1}{z}):\mathbb{Q}]=\frac{1}{2}\varphi(n)$ for every primitive $n$-th root of unity $z$

Published on 10 May 2026 - 22:04

List Question

Do the zeroes of this polynomial lie inside, outside, or on the unit circle? $P_n(z)=1^3 z + 2^3 z^2 + 3^3 z^3 + \cdots + n^3 z^n$

What is the sum of the squares of the 10th roots of unity?

If $y^4 = 5$ and $z^6 = 15$, then $y \notin \mathbb{Q}(z)$

Determine a Normal Basis for Galois Extension of $\mathbb{Q}$ with primitive pth root of unit (p prime)

Finding the Normal Basis of Cyclotomic field

Why is the magnitude of the sum of two adjacent nth roots always an 'interesting' number, and what do these numbers have to do with each other?

How to get common roots of unity of $ z^{6}=1 $ and $ z^{21}=1 $?

$\zeta_n^k$ is primitive if and only if $(k,n) = 1$

$ \bigcup_{n=1}^{\infty}\{z\in\Bbb C:z^n=1\}=\{z\in\Bbb C:|z|=1\}$?

Why is $\frac{\cos(\alpha + \beta) + \cos(-\alpha)}{\sin(\alpha + \beta) + \sin(-\alpha)}$ independent of $\alpha$?

(Degree of) Splitting Field for $f(x) = x^p - 2$ over $\mathbb{Q}$, p prime

Prove that $z_1,z_2,z_3$ with equal, non-zero modulus, are vertices of an equilateral triangle if and only if $\sum_{cyc}|z_1-z_2|(z_1+z_2) = 0 $

Kummer ring - special monic polynomial with zero at root of unity

Why are roots of unity evenly spaced?

Let $n>2$ prove that $[\mathbb Q(z+\frac{1}{z}):\mathbb{Q}]=\frac{1}{2}\varphi(n)$ for every primitive $n$-th root of unity $z$

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