d) ej Z [√2] is a field Z [√2] is an integral domain
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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Number 2 d and e
![1.
2 is an integral domain, but not a field.
Z, is a field for every prime number p.
2.
EXERCISE:
1.
Let <A, +,.> be a ring with unity and let A* be the set of all units in A. Show that
the set <A*,.> is a group.
2.
Let Z [√2] = { a+b√2 labe Z}. Define addition and multiplication on
Z[√2] as follows:
(a+b√2)+(c + d√2) (a + c)+(b+d)√2
(a+b√2)(c + d√2)
= (ac+2bd) + (ad + bc)√2
Prove or disprove the following:
a)
Z [√2] is a ring
b)
Z [√2] is a commutative ring
Z [√2] is a ring with unity
Z [√2] is a field
Z [√2] is an integral domain
Show that S = {(a,a) | ae Z) is a subring of Zx Z (Use the definition of
addition and multiplication in direct products)
Determine whether T = {{ (a,- a) | ae Z) is a subring of Zx Z
7
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Transcribed Image Text:1.
2 is an integral domain, but not a field.
Z, is a field for every prime number p.
2.
EXERCISE:
1.
Let <A, +,.> be a ring with unity and let A* be the set of all units in A. Show that
the set <A*,.> is a group.
2.
Let Z [√2] = { a+b√2 labe Z}. Define addition and multiplication on
Z[√2] as follows:
(a+b√2)+(c + d√2) (a + c)+(b+d)√2
(a+b√2)(c + d√2)
= (ac+2bd) + (ad + bc)√2
Prove or disprove the following:
a)
Z [√2] is a ring
b)
Z [√2] is a commutative ring
Z [√2] is a ring with unity
Z [√2] is a field
Z [√2] is an integral domain
Show that S = {(a,a) | ae Z) is a subring of Zx Z (Use the definition of
addition and multiplication in direct products)
Determine whether T = {{ (a,- a) | ae Z) is a subring of Zx Z
7
b)
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