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 796 · Answer 1 · 2019-01-05 09:54:32

796

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 05 Jan 2019 - 9:54

score 1.6k · Answer 2 · 2019-01-09 16:17:58

1.6k

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

Published on 09 Jan 2019 - 16:17

score 84 · Answer 3 · 2019-01-11 00:45:08

84

Views

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

Published on 11 Jan 2019 - 0:45

score 188 · Answer 4 · 2026-03-25 10:35:30

188

Views

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

Published on 25 Mar 2026 - 10:35

score 455 · Answer 5 · 2026-03-25 01:23:56

455

Views

Finding the Normal Basis of Cyclotomic field

Published on 25 Mar 2026 - 1:23

score 163 · Answer 6 · 2019-01-18 02:28:48

163

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 18 Jan 2019 - 2:28

score 235 · Answer 7 · 2019-01-23 00:43:49

235

Views

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

Published on 23 Jan 2019 - 0:43

score 194 · Answer 8 · 2026-03-25 13:33:17

194

Views

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

Published on 25 Mar 2026 - 13:33

score 1k · Answer 9 · 2019-02-01 15:14:47

1k

Views

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

Published on 01 Feb 2019 - 15:14

score 86 · Answer 10 · 2019-02-06 22:34:52

86

Views

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

Published on 06 Feb 2019 - 22:34

score 2.1k · Answer 11 · 2026-03-25 22:30:42

2.1k

Views

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

Published on 25 Mar 2026 - 22:30

score 106 · Answer 12 · 2019-02-12 12:38:58

106

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 12 Feb 2019 - 12:38

score 89 · Answer 13 · 2019-02-12 20:43:54

89

Views

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

Published on 12 Feb 2019 - 20:43

score 1.3k · Answer 14 · 2019-02-13 07:06:19

1.3k

Views

Why are roots of unity evenly spaced?

Published on 13 Feb 2019 - 7:06

score 50 · Answer 15 · 2026-03-25 19:06:18

50

Views

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 25 Mar 2026 - 19:06

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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