Question

$x^2 + y^2 + z^2 - 2xy - 2xz - 2yz$ is irreducible over $\mathbb{Z}[x, y, z]$

score 126 · Answer 1 · 2023-10-12 20:51:14

126

Views

$x^2 + y^2 + z^2 - 2xy - 2xz - 2yz$ is irreducible over $\mathbb{Z}[x, y, z]$

Published on 12 Oct 2023 - 20:51

score 67 · Answer 2 · 2023-10-29 15:55:55

67

Views

If $R$ is not a UFD then $RS^{-1}$ is not a field

Published on 29 Oct 2023 - 15:55

score 58 · Answer 3 · 2026-03-25 22:04:20

58

Views

What is the factorization of 26 in $\mathbb Z[i]$ into irreducible?

Published on 25 Mar 2026 - 22:04

score 38 · Answer 4 · 2023-11-10 17:53:33

38

Views

How can I compare the number of irreducible factors of one element with another?

Published on 10 Nov 2023 - 17:53

score 97 · Answer 5 · 2026-03-28 08:35:33

97

Views

Exhibit all the ideals in the ring $F[x]/(p(x))$, where $F$ is a field and $p(x)∈F[x]$ (describe them in terms of the factorization of $p(x)$)

Published on 28 Mar 2026 - 8:35

score 50 · Answer 6 · 2023-11-15 00:57:25

50

Views

If every irreducible elements in Noetherian ring is prime, it is UFD

Published on 15 Nov 2023 - 0:57

score 91 · Answer 7 · 2023-11-27 16:21:56

91

Views

Primitive $g\mid af\Rightarrow g\mid f,\,$ for $\,0\!\neq\! a\in $ UFD A, $\,f,g\in A[x],\,$ by Gauss's Lemma

Published on 27 Nov 2023 - 16:21

score 62 · Answer 8 · 2023-12-04 17:29:14

62

Views

Proof about gcd and resultant in UFD

Published on 04 Dec 2023 - 17:29

score 61 · Answer 9 · 2023-12-21 18:35:46

61

Views

Let R be a UFD such that each maximal ideal of R is a principal ideal, prove that R is a PID

Published on 21 Dec 2023 - 18:35

score 61 · Answer 10 · 2023-12-23 11:37:22

61

Views

Let $R$ be a UFD, and let $S$ be a multiplicative subset of $R$ containing the unity of $R$ then $R_S$ is also UFD

Published on 23 Dec 2023 - 11:37

score 112 · Answer 11 · 2023-12-30 22:34:08

112

Views

Unique factorization domain property in $\mathbb{Z[\mathcal{i}]}$

Published on 30 Dec 2023 - 22:34

score 31 · Answer 12 · 2024-01-09 16:32:14

31

Views

Factorization into irreducibles in $\mathbb{Z}[i\sqrt{n}]$ with square-free $n \in \mathbb{Z}^{+}$

Published on 09 Jan 2024 - 16:32

score 91 · Answer 13 · 2024-01-09 20:20:35

91

Views

Factorize $x^4 + 4x^2 +1$ over $\mathbb{F}_{29}$

Published on 09 Jan 2024 - 20:20

score 78 · Answer 14 · 2024-02-03 12:03:56

78

Views

Explicit proof of the fact that a non integrally closed domain is not a UFD

Published on 03 Feb 2024 - 12:03

score 77 · Answer 15 · 2024-02-06 03:09:07

77

Views

Need help understanding the proof that if $R$ is a PID, every nonzero, nonunit elements of $R$ is a product of irreducibles.

Published on 06 Feb 2024 - 3:09

List Question

$x^2 + y^2 + z^2 - 2xy - 2xz - 2yz$ is irreducible over $\mathbb{Z}[x, y, z]$

If $R$ is not a UFD then $RS^{-1}$ is not a field

What is the factorization of 26 in $\mathbb Z[i]$ into irreducible?

How can I compare the number of irreducible factors of one element with another?

Exhibit all the ideals in the ring $F[x]/(p(x))$, where $F$ is a field and $p(x)∈F[x]$ (describe them in terms of the factorization of $p(x)$)

If every irreducible elements in Noetherian ring is prime, it is UFD

Primitive $g\mid af\Rightarrow g\mid f,\,$ for $\,0\!\neq\! a\in $ UFD A, $\,f,g\in A[x],\,$ by Gauss's Lemma

Proof about gcd and resultant in UFD

Let R be a UFD such that each maximal ideal of R is a principal ideal, prove that R is a PID

Let $R$ be a UFD, and let $S$ be a multiplicative subset of $R$ containing the unity of $R$ then $R_S$ is also UFD

Unique factorization domain property in $\mathbb{Z[\mathcal{i}]}$

Factorization into irreducibles in $\mathbb{Z}[i\sqrt{n}]$ with square-free $n \in \mathbb{Z}^{+}$

Factorize $x^4 + 4x^2 +1$ over $\mathbb{F}_{29}$

Explicit proof of the fact that a non integrally closed domain is not a UFD

Need help understanding the proof that if $R$ is a PID, every nonzero, nonunit elements of $R$ is a product of irreducibles.

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