2. Let Z[/2] = {a+b2 |a, b eZ} and let H= { a 2b : a,b eZ}. a. Show that Z[2] and H are isomorphic as rings.
2. Let Z[/2] = {a+b2 |a, b eZ} and let H= { a 2b : a,b eZ}. a. Show that Z[2] and H are isomorphic as rings.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
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I need question #2 please
![**Math 5220 Homework #6**
*The following problems are based on lectures #15 – 16.*
1. Let \( S = \left\{ \begin{bmatrix} a & b \\ -b & a \end{bmatrix} : a, b \in \mathbb{R} \right\} \). Show that \( \phi : \mathbb{C} \rightarrow S \) defined by
\[
\phi(a + bi) = \begin{bmatrix} a & b \\ -b & a \end{bmatrix}
\]
is a ring homomorphism.
2. Let \( \mathbb{Z}[\sqrt{2}] = \{ a + b\sqrt{2} \mid a, b \in \mathbb{Z} \} \) and let \( H = \left\{ \begin{bmatrix} a & 2b \\ b & a \end{bmatrix} : a, b \in \mathbb{Z} \right\} \).
Show that \( \mathbb{Z}[\sqrt{2}] \) and \( H \) are isomorphic as rings.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fcfc5bdab-6654-444e-b9d5-5fc4b8470fbb%2Ff0687134-1640-4dea-8686-c533c7f7e4c5%2Flbnjkcp.jpeg&w=3840&q=75)
Transcribed Image Text:**Math 5220 Homework #6**
*The following problems are based on lectures #15 – 16.*
1. Let \( S = \left\{ \begin{bmatrix} a & b \\ -b & a \end{bmatrix} : a, b \in \mathbb{R} \right\} \). Show that \( \phi : \mathbb{C} \rightarrow S \) defined by
\[
\phi(a + bi) = \begin{bmatrix} a & b \\ -b & a \end{bmatrix}
\]
is a ring homomorphism.
2. Let \( \mathbb{Z}[\sqrt{2}] = \{ a + b\sqrt{2} \mid a, b \in \mathbb{Z} \} \) and let \( H = \left\{ \begin{bmatrix} a & 2b \\ b & a \end{bmatrix} : a, b \in \mathbb{Z} \right\} \).
Show that \( \mathbb{Z}[\sqrt{2}] \) and \( H \) are isomorphic as rings.
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