Answered: 7. a) Prove that every field is a…

7. a) Prove that every field is a principal ideal ring. b) Consider the set of numbers R = {a+ bv2|a, b€ Z}. Show that the ring (R, +,) is not a field by exhibiting a nontrivial ideal of (R, +, ).

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

Verify that (Z1, +,) is a field if and only if (R, +, ) has positive characteristics.
7. a) Prove that every field is a principal ideal ring.
b) Consider the set of numbers R (a+ bv2|a, bE 2}. Show that the ring
(R, +,) is not a field by exhibiting a nontrivial ideal of (R,+, ).

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