Question 6 Consider two groups Q[x] (defined in Question 3(v)) and Q, both under the usual + of polynomials and numbers, respectively. Prove that these two algebraic structures (groups?) are not isomorphic. (Hint: Note these structures happen to also be Q-vector spaces under the right choice of operations, and these spaces have different dimension. However, you cannot appeal to linear algebra, because we are not viewing these objects as vector spaces, but maybe this will give you an idea. Think of some property that holds in, e.g., Q but not in Q[x] or vice versa, and so that this property uses only the group operation in its description and is independent of notation. )

Question 6 Consider two groups Q[x] (defined in Question 3(v)) and Q, both under the usual + of polynomials and numbers, respectively. Prove that these two algebraic structures (groups?) are not isomorphic. (Hint: Note these structures happen to also be Q-vector spaces under the right choice of operations, and these spaces have different dimension. However, you cannot appeal to linear algebra, because we are not viewing these objects as vector spaces, but maybe this will give you an idea. Think of some property that holds in, e.g., Q but not in Q[x] or vice versa, and so that this property uses only the group operation in its description and is independent of notation. )

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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