Question

Showing the $q$-th cyclotomic polynomial is irreducible over $\mathbb{F}_p$ if $p$ is a primitive root mod $q$

score 1.3k · Answer 1 · 2017-05-13 01:56:15

1.3k

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Showing the $q$-th cyclotomic polynomial is irreducible over $\mathbb{F}_p$ if $p$ is a primitive root mod $q$

Published on 13 May 2017 - 1:56

score 2.2k · Answer 2 · 2017-05-18 16:28:33

2.2k

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If $K/\mathbb{Q}$ is finite, then $K$ contains finitely many $n$th roots of unity

Published on 18 May 2017 - 16:28

score 252 · Answer 3 · 2017-05-21 02:03:02

252

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How can I see $\Phi_{p^r}(x)=(x^{p^{r-1}})^{p-1}+(x^{p^{r-1}})^{p-2}+\cdots+(x^{p^{r-1}})+1$ and is irreducible?

Published on 21 May 2017 - 2:03

score 423 · Answer 4 · 2026-03-25 23:53:13

423

Views

Question about the proof that cyclotomic polynomials are irreducible.

Published on 25 Mar 2026 - 23:53

score 693 · Answer 5 · 2026-03-25 23:56:30

693

Views

Relationship between the norm and the minimal polynomial

Published on 25 Mar 2026 - 23:56

score 74 · Answer 6 · 2017-06-02 10:37:41

74

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Sum of prime values of Cyclotomic Polynomials of an integer evaluated at an integer

Published on 02 Jun 2017 - 10:37

score 61 · Answer 7 · 2017-06-10 09:15:47

61

Views

Degree of splitting field of $\big|1+e^{\frac{1}{11}2\pi i}+e^{\frac{3}{11}2\pi i}\big|$ over $\mathbb Q$

Published on 10 Jun 2017 - 9:15

score 1.2k · Answer 8 · 2017-07-04 12:26:53

1.2k

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How existence of an irreducible polynomial in $F_p(x)$ of degree $n$ guarantees existence of finite field of order $p^n$

Published on 04 Jul 2017 - 12:26

score 28 · Answer 9 · 2026-03-26 01:28:48

28

Views

$p$ multi-variate polynomial over Finite Field for prime $p$.

Published on 26 Mar 2026 - 1:28

score 65 · Answer 10 · 2017-07-12 20:18:30

65

Views

Simplifying $\text{lcm}(a^{n/p_1}-1, a^{n/p_2}-1,\dots, a^{n/p_m}-1)$

Published on 12 Jul 2017 - 20:18

score 412 · Answer 11 · 2017-07-19 11:23:18

412

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Common factors of cyclotomic polynomials and polynomials with prime coefficients

Published on 19 Jul 2017 - 11:23

score 359 · Answer 12 · 2026-03-26 01:26:56

359

Views

Cyclotomic Extension over Finite Fields

Published on 26 Mar 2026 - 1:26

score 118 · Answer 13 · 2017-09-06 10:11:34

118

Views

Polynomial ring $\mathbb{Z}[X]/(X^{2^k} + 1)$ has property $X^{2^k} = −1$?

Published on 06 Sep 2017 - 10:11

score 113 · Answer 14 · 2017-09-10 21:35:28

113

Views

A kind of GCD of cyclotomic polynomials

Published on 10 Sep 2017 - 21:35

score 213 · Answer 15 · 2017-09-22 14:35:06

213

Views

If $d$ is the order of $a$ modulo $n$ then prove that $n| \Phi_d(a)$

Published on 22 Sep 2017 - 14:35

List Question

Showing the $q$-th cyclotomic polynomial is irreducible over $\mathbb{F}_p$ if $p$ is a primitive root mod $q$

If $K/\mathbb{Q}$ is finite, then $K$ contains finitely many $n$th roots of unity

How can I see $\Phi_{p^r}(x)=(x^{p^{r-1}})^{p-1}+(x^{p^{r-1}})^{p-2}+\cdots+(x^{p^{r-1}})+1$ and is irreducible?

Question about the proof that cyclotomic polynomials are irreducible.

Relationship between the norm and the minimal polynomial

Sum of prime values of Cyclotomic Polynomials of an integer evaluated at an integer

Degree of splitting field of $\big|1+e^{\frac{1}{11}2\pi i}+e^{\frac{3}{11}2\pi i}\big|$ over $\mathbb Q$

How existence of an irreducible polynomial in $F_p(x)$ of degree $n$ guarantees existence of finite field of order $p^n$

$p$ multi-variate polynomial over Finite Field for prime $p$.

Simplifying $\text{lcm}(a^{n/p_1}-1, a^{n/p_2}-1,\dots, a^{n/p_m}-1)$

Common factors of cyclotomic polynomials and polynomials with prime coefficients

Cyclotomic Extension over Finite Fields

Polynomial ring $\mathbb{Z}[X]/(X^{2^k} + 1)$ has property $X^{2^k} = −1$?

A kind of GCD of cyclotomic polynomials

If $d$ is the order of $a$ modulo $n$ then prove that $n| \Phi_d(a)$

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