Question

Let $K = \mathbb{Q}\left(\frac{-1+\sqrt{-3}}{2}\right)$. Show that $\mathcal{O}_K = \mathbb{Z}\left[\frac{-1+\sqrt{-3}}{2}\right]$.

score 149 · Answer 1 · 2018-01-22 15:50:24

149

Views

Let $K = \mathbb{Q}\left(\frac{-1+\sqrt{-3}}{2}\right)$. Show that $\mathcal{O}_K = \mathbb{Z}\left[\frac{-1+\sqrt{-3}}{2}\right]$.

Published on 22 Jan 2018 - 15:50

score 2.2k · Answer 2 · 2018-01-22 17:28:57

2.2k

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Set closed under multiplication

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score 894 · Answer 3 · 2018-01-22 17:59:57

894

Views

Show that in the ring of quaternions every non-zero element has a multiplicative inverse

Published on 22 Jan 2018 - 17:59

score 313 · Answer 4 · 2018-01-22 19:32:59

313

Views

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score 75 · Answer 5 · 2026-03-12 10:27:41

75

Views

Show that $\Bbb Z\Big(\frac{1+\sqrt{d}}{2}\Big)$ is closed under multiplication.

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score 87 · Answer 6 · 2026-03-16 16:13:13

87

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Module and Ring

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score 1.9k · Answer 7 · 2026-03-16 20:06:20

1.9k

Views

Prove that an idempotent element must be either 0, 1 or a zero-divisor.

Published on 16 Mar 2026 - 20:06

score 110 · Answer 8 · 2018-01-23 06:06:50

110

Views

Identify $\mathbb{Z}[x]/(x^2+3, 5)$

Published on 23 Jan 2018 - 6:06

score 127 · Answer 9 · 2026-03-17 07:38:07

127

Views

Zero intersection of two ideals in the ring $R=\mathbb{Z}_{4}\left[ x\right] /\left\langle x^{n}-1\right\rangle .$

Published on 17 Mar 2026 - 7:38

score 304 · Answer 10 · 2018-01-23 12:26:38

304

Views

splitting field for a finite dimensional central simple F-algebra$A$

Published on 23 Jan 2018 - 12:26

score 166 · Answer 11 · 2026-03-22 04:16:37

166

Views

A $G$-module admits a surjection from a $G$-module, which is free as an abelian group, such that the kernel is free

Published on 22 Mar 2026 - 4:16

score 1k · Answer 12 · 2026-03-21 18:28:04

1k

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Nilradical of a noncommutative Ring

Published on 21 Mar 2026 - 18:28

score 68 · Answer 13 · 2018-01-23 16:12:38

68

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Polynomial division in $\mathbb{Z}_5[x]$

Published on 23 Jan 2018 - 16:12

score 145 · Answer 14 · 2018-01-23 19:50:02

145

Views

Quasi-Coherence of $Hom$ Sheaf

Published on 23 Jan 2018 - 19:50

score 52 · Answer 15 · 2018-01-23 23:19:09

52

Views

Properties of Ring Completion

Published on 23 Jan 2018 - 23:19

List Question

Let $K = \mathbb{Q}\left(\frac{-1+\sqrt{-3}}{2}\right)$. Show that $\mathcal{O}_K = \mathbb{Z}\left[\frac{-1+\sqrt{-3}}{2}\right]$.

Set closed under multiplication

Show that in the ring of quaternions every non-zero element has a multiplicative inverse

Difficulties with constructing a surjective ring homomorphism

Show that $\Bbb Z\Big(\frac{1+\sqrt{d}}{2}\Big)$ is closed under multiplication.

Module and Ring

Prove that an idempotent element must be either 0, 1 or a zero-divisor.

Identify $\mathbb{Z}[x]/(x^2+3, 5)$

Zero intersection of two ideals in the ring $R=\mathbb{Z}_{4}\left[ x\right] /\left\langle x^{n}-1\right\rangle .$

splitting field for a finite dimensional central simple F-algebra$A$

A $G$-module admits a surjection from a $G$-module, which is free as an abelian group, such that the kernel is free

Nilradical of a noncommutative Ring

Polynomial division in $\mathbb{Z}_5[x]$

Quasi-Coherence of $Hom$ Sheaf

Properties of Ring Completion

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