Question

Why is $\mathbb{Z}_3[x]/\langle x^2+1\rangle$ a field with order $9$? And does order $9$ mean it has $9$ elements?

score 811 · Answer 1 · 2019-02-23 20:48:37

811

Views

Why is $\mathbb{Z}_3[x]/\langle x^2+1\rangle$ a field with order $9$? And does order $9$ mean it has $9$ elements?

Published on 23 Feb 2019 - 20:48

score 69 · Answer 2 · 2026-03-26 12:58:30

69

Views

Does this algebraic structure belong to a subclass of neofield (or other structure)?

Published on 26 Mar 2026 - 12:58

score 581 · Answer 3 · 2026-03-27 00:09:43

581

Views

Finding a field extension in which every element has zero trace

Published on 27 Mar 2026 - 0:09

score 34 · Answer 4 · 2019-02-26 17:32:30

34

Views

Find an element that generates the Ideal

Published on 26 Feb 2019 - 17:32

score 1k · Answer 5 · 2026-03-25 22:26:51

1k

Views

$f(x)$ is irreducible over finite field $F_p$ , does $f(x)$ split over $F_p / \langle f(x) \rangle$?

Published on 25 Mar 2026 - 22:26

score 119 · Answer 6 · 2019-02-27 09:31:24

119

Views

A $5\times 5$ Matrix with No Eigenvalues.

Published on 27 Feb 2019 - 9:31

score 344 · Answer 7 · 2026-03-26 08:04:04

344

Views

Extended Hamming code to cyclic code

Published on 26 Mar 2026 - 8:04

score 39 · Answer 8 · 2026-03-24 20:32:03

39

Views

Estimate for the genus of a global function field

Published on 24 Mar 2026 - 20:32

score 42 · Answer 9 · 2019-02-28 11:26:04

42

Views

Primitive elements of finite field

Published on 28 Feb 2019 - 11:26

score 223 · Answer 10 · 2026-03-26 07:56:54

223

Views

Relation between degree and Hamming distance

Published on 26 Mar 2026 - 7:56

score 72 · Answer 11 · 2019-03-01 09:38:32

72

Views

Frobenius morphism associated to field extension

Published on 01 Mar 2019 - 9:38

score 53 · Answer 12 · 2019-03-01 11:42:38

53

Views

Root of irreducible polynomial not in general a generator

Published on 01 Mar 2019 - 11:42

score 54 · Answer 13 · 2026-03-25 23:13:09

54

Views

Can $p(x)\in \mathbb{F}_{3}(x)$ with $p(x)=\frac{x²+x+1}{x+1}$ be expressed as a polynomial? Is it not possible for any of the given fields?

Published on 25 Mar 2026 - 23:13

score 105 · Answer 14 · 2019-03-01 17:23:39

105

Views

Difference between splitting field of $X^{15}-2$ and $X^{15}-1$ over finite field $\mathbb{F}_7$

Published on 01 Mar 2019 - 17:23

score 1.8k · Answer 15 · 2026-03-25 22:26:54

1.8k

Views

Degree of splitting field of $X^n-1$ over some finite field

Published on 25 Mar 2026 - 22:26

List Question

Why is $\mathbb{Z}_3[x]/\langle x^2+1\rangle$ a field with order $9$? And does order $9$ mean it has $9$ elements?

Does this algebraic structure belong to a subclass of neofield (or other structure)?

Finding a field extension in which every element has zero trace

Find an element that generates the Ideal

$f(x)$ is irreducible over finite field $F_p$ , does $f(x)$ split over $F_p / \langle f(x) \rangle$?

A $5\times 5$ Matrix with No Eigenvalues.

Extended Hamming code to cyclic code

Estimate for the genus of a global function field

Primitive elements of finite field

Relation between degree and Hamming distance

Frobenius morphism associated to field extension

Root of irreducible polynomial not in general a generator

Can $p(x)\in \mathbb{F}_{3}(x)$ with $p(x)=\frac{x²+x+1}{x+1}$ be expressed as a polynomial? Is it not possible for any of the given fields?

Difference between splitting field of $X^{15}-2$ and $X^{15}-1$ over finite field $\mathbb{F}_7$

Degree of splitting field of $X^n-1$ over some finite field

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