Question

If $3$ divides $p^n - 1$ does $\mathbf{F}_{p^n}$ contain a primitive third root of unity?

score 129 · Answer 1 · 2016-08-15 23:26:36

129

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If $3$ divides $p^n - 1$ does $\mathbf{F}_{p^n}$ contain a primitive third root of unity?

Published on 15 Aug 2016 - 23:26

score 302 · Answer 2 · 2026-03-25 13:34:19

302

Views

Determine all $P(X)\in K[X]$ such that $P\big(X^2+1\big)=\big(P(X)\big)^2+1$, for fields $K$ of any characteristic.

Published on 25 Mar 2026 - 13:34

score 2.9k · Answer 3 · 2016-08-22 13:57:46

2.9k

Views

Find the Galois group of $x^3-2$ over the field $\mathbb F_5$

Published on 22 Aug 2016 - 13:57

score 1.6k · Answer 4 · 2026-03-29 18:15:19

1.6k

Views

$\mathbb F_q(x)$, the field of rational functions over a finite field, is not finitely generated as a ring.

Published on 29 Mar 2026 - 18:15

score 96 · Answer 5 · 2016-08-22 21:00:48

96

Views

Show that if $char(\mathbb{F})$ is prime, then $\forall k \in \mathbb{N}, \forall a, b \in \mathbb{F}: (a+b)^{p^k} = a^{p^k} + b^{p^k}$

Published on 22 Aug 2016 - 21:00

score 103 · Answer 6 · 2016-08-24 05:08:13

103

Views

How can I factor the polynomial $9x^4+16$ in $\mathbb{F}_{103}$?

Published on 24 Aug 2016 - 5:08

score 265 · Answer 7 · 2016-08-25 21:18:02

265

Views

Low degree polynomials over finite fields with no roots

Published on 25 Aug 2016 - 21:18

score 75 · Answer 8 · 2016-08-26 09:18:19

75

Views

Probability that an arbitrary element of a field equals a uniformly random value

Published on 26 Aug 2016 - 9:18

score 83 · Answer 9 · 2016-08-26 20:18:00

83

Views

Given a finite field F, is there a nice way to construct a finite field of pairs of elements of F?

Published on 26 Aug 2016 - 20:18

score 104 · Answer 10 · 2026-03-26 11:02:13

104

Views

Degree of $\mathbb{F}_p(\zeta_r)$ over $\mathbb{F}_p$

Published on 26 Mar 2026 - 11:02

score 384 · Answer 11 · 2016-08-27 11:54:23

384

Views

Every finite division ring has order $|Z|^n$ for some $n\in\Bbb N$ and where $Z$ is the center.

Published on 27 Aug 2016 - 11:54

score 229 · Answer 12 · 2016-08-27 18:45:14

229

Views

Lottery System vs combinatorics and geometry

Published on 27 Aug 2016 - 18:45

score 365 · Answer 13 · 2016-08-28 21:34:37

365

Views

Why aren't all prime subfields equal to {0,1}?

Published on 28 Aug 2016 - 21:34

score 128 · Answer 14 · 2026-03-25 16:39:27

128

Views

Time complexity of Jacobi sums

Published on 25 Mar 2026 - 16:39

score 1.7k · Answer 15 · 2016-08-29 18:58:06

1.7k

Views

Is there a clever(er) way to construct a finite field with nine elements?

Published on 29 Aug 2016 - 18:58

List Question

If $3$ divides $p^n - 1$ does $\mathbf{F}_{p^n}$ contain a primitive third root of unity?

Determine all $P(X)\in K[X]$ such that $P\big(X^2+1\big)=\big(P(X)\big)^2+1$, for fields $K$ of any characteristic.

Find the Galois group of $x^3-2$ over the field $\mathbb F_5$

$\mathbb F_q(x)$, the field of rational functions over a finite field, is not finitely generated as a ring.

Show that if $char(\mathbb{F})$ is prime, then $\forall k \in \mathbb{N}, \forall a, b \in \mathbb{F}: (a+b)^{p^k} = a^{p^k} + b^{p^k}$

How can I factor the polynomial $9x^4+16$ in $\mathbb{F}_{103}$?

Low degree polynomials over finite fields with no roots

Probability that an arbitrary element of a field equals a uniformly random value

Given a finite field F, is there a nice way to construct a finite field of pairs of elements of F?

Degree of $\mathbb{F}_p(\zeta_r)$ over $\mathbb{F}_p$

Every finite division ring has order $|Z|^n$ for some $n\in\Bbb N$ and where $Z$ is the center.

Lottery System vs combinatorics and geometry

Why aren't all prime subfields equal to {0,1}?

Time complexity of Jacobi sums

Is there a clever(er) way to construct a finite field with nine elements?

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