1. Find all units in Z[V-3]. Justify your answer. 2. Prove that 2 is an irreducible element in Z[V-3]. 3. * Let Z[V3) = {a+ bv3: a,be Z} be the subring of the ring of complex numbers. Show that there are infinitely many units in Z[v3).3
1. Find all units in Z[V-3]. Justify your answer. 2. Prove that 2 is an irreducible element in Z[V-3]. 3. * Let Z[V3) = {a+ bv3: a,be Z} be the subring of the ring of complex numbers. Show that there are infinitely many units in Z[v3).3
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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![This homework is related to Gaussian integers. This is our last homework. Thank
you for preparing your solutions!
We define Z[V-3] = {a + biv3: a, b e Z} to be the subring of the ring of complex
numbers, where as usual i2 = -1.
As per Gaussian integers, we say that a divides b in Z[V-3) if there exists e e Z[V-3)
such that ab= c.
If c divides 1 in Z[V=3] then c is called a unit in Z[V-3].
An element r in Z[/-3] is called irreducible if it is not a product of two non units in
ZV-3).
An element a e Z[V-3] is called prime if whenever a divides be for some b, e e Z[V-3]
then either a divides b in Z[V-3] or a divides c in Z[V=3.
1. Find all units in Z[V-3]. Justify your answer.
2. Prove that 2 is an irreducible element in Z[V-3].
3. * Let Z[V3 = {a+bv3: a,be Z} be the subring of the ring of complex numbers.
Show that there are infinitely many units in Z[V3).:](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F52a0a97f-79dc-47c8-8231-1e20a340488c%2F7a37d07e-dfdf-43c0-ae77-991967926500%2Frxkogmg_processed.jpeg&w=3840&q=75)
Transcribed Image Text:This homework is related to Gaussian integers. This is our last homework. Thank
you for preparing your solutions!
We define Z[V-3] = {a + biv3: a, b e Z} to be the subring of the ring of complex
numbers, where as usual i2 = -1.
As per Gaussian integers, we say that a divides b in Z[V-3) if there exists e e Z[V-3)
such that ab= c.
If c divides 1 in Z[V=3] then c is called a unit in Z[V-3].
An element r in Z[/-3] is called irreducible if it is not a product of two non units in
ZV-3).
An element a e Z[V-3] is called prime if whenever a divides be for some b, e e Z[V-3]
then either a divides b in Z[V-3] or a divides c in Z[V=3.
1. Find all units in Z[V-3]. Justify your answer.
2. Prove that 2 is an irreducible element in Z[V-3].
3. * Let Z[V3 = {a+bv3: a,be Z} be the subring of the ring of complex numbers.
Show that there are infinitely many units in Z[V3).:
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