Let R be the ring Z[/2] = {a + bv2|a,b e Z}. For each of the following elements of R determine if it is an invertible element of R, an irreducible element of R, or neither invertible nor irreducible: (i) V2, (ii) 1+ v2, (iii) 2+ v2, (iv) 3+ v2, (v) 4 + v2, (vi) 5+ v2.

Let R be the ring Z[/2] = {a + bv2|a,b e Z}. For each of the following elements of R determine if it is an invertible element of R, an irreducible element of R, or neither invertible nor irreducible: (i) V2, (ii) 1+ v2, (iii) 2+ v2, (iv) 3+ v2, (v) 4 + v2, (vi) 5+ v2.

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

Let R be the ring Z[V2] = {a + b/2|a, b € Z}. For each of the following elements of R
determine if it is an invertible element of R, an irreducible element of R, or neither invertible
nor irreducible:
(i) V2,
(ii) 1+ v2, (iii) 2 + v2,
(iv) 3 + v2, (v) 4 + v2, (vi) 5 + V2.
Hint: Use the multiplicativity of the function R → Z that sends m + n/2 to m² – 2n².

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