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
Related questions
Question
100%
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
b)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F9949b50e-c59e-4a5e-95ac-404b1c8ac983%2F32087fa6-056f-49f5-be2a-e65e59be64ff%2Ffusg99_processed.jpeg&w=3840&q=75)
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)
Expert Solution
![](/static/compass_v2/shared-icons/check-mark.png)
This question has been solved!
Explore an expertly crafted, step-by-step solution for a thorough understanding of key concepts.
Step by step
Solved in 2 steps with 2 images
![Blurred answer](/static/compass_v2/solution-images/blurred-answer.jpg)
Recommended textbooks for you
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
![Introductory Mathematics for Engineering Applicat…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Advanced Engineering Mathematics](https://www.bartleby.com/isbn_cover_images/9780470458365/9780470458365_smallCoverImage.gif)
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
![Numerical Methods for Engineers](https://www.bartleby.com/isbn_cover_images/9780073397924/9780073397924_smallCoverImage.gif)
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
![Introductory Mathematics for Engineering Applicat…](https://www.bartleby.com/isbn_cover_images/9781118141809/9781118141809_smallCoverImage.gif)
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
![Mathematics For Machine Technology](https://www.bartleby.com/isbn_cover_images/9781337798310/9781337798310_smallCoverImage.jpg)
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
![Basic Technical Mathematics](https://www.bartleby.com/isbn_cover_images/9780134437705/9780134437705_smallCoverImage.gif)
![Topology](https://www.bartleby.com/isbn_cover_images/9780134689517/9780134689517_smallCoverImage.gif)