2. Let s-{(소)아 2a S = 26 2a Assume that S is a subring of M2(Q). Prove that S is isomorphic to Q[V2] as follows: Let f: SQ[/2] be defined by )) 2a = 2a + by2. 26 2a (a) Prove that f is a ring homomorphism. (b) Prove that f is bijective.
2. Let s-{(소)아 2a S = 26 2a Assume that S is a subring of M2(Q). Prove that S is isomorphic to Q[V2] as follows: Let f: SQ[/2] be defined by )) 2a = 2a + by2. 26 2a (a) Prove that f is a ring homomorphism. (b) Prove that f is bijective.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Question
absract algebra
![**Let**
\[ S = \left\{ \begin{pmatrix} 2a & b \\ 2b & 2a \end{pmatrix} : a, b \in \mathbb{Q} \right\}. \]
Assume that \( S \) is a subring of \( M_2(\mathbb{Q}) \). Prove that \( S \) is isomorphic to \( \mathbb{Q}[\sqrt{2}] \) as follows:
Let \( f: S \to \mathbb{Q}[\sqrt{2}] \) be defined by
\[ f \left( \begin{pmatrix} 2a & b \\ 2b & 2a \end{pmatrix} \right) = 2a + b\sqrt{2}. \]
(a) Prove that \( f \) is a ring homomorphism.
(b) Prove that \( f \) is bijective.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d940ce8-cba2-4a95-af25-aae0739ca5aa%2F52c51830-e003-458d-a71c-748d15035f6d%2F6kc71ub_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Let**
\[ S = \left\{ \begin{pmatrix} 2a & b \\ 2b & 2a \end{pmatrix} : a, b \in \mathbb{Q} \right\}. \]
Assume that \( S \) is a subring of \( M_2(\mathbb{Q}) \). Prove that \( S \) is isomorphic to \( \mathbb{Q}[\sqrt{2}] \) as follows:
Let \( f: S \to \mathbb{Q}[\sqrt{2}] \) be defined by
\[ f \left( \begin{pmatrix} 2a & b \\ 2b & 2a \end{pmatrix} \right) = 2a + b\sqrt{2}. \]
(a) Prove that \( f \) is a ring homomorphism.
(b) Prove that \( f \) is bijective.
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