Now let ap be a nonzero element in Z/pZ. Show that a, is invertible. (Hint: Invoking the fact that a and p are relatively prime, we find that there must exist integers x and y such that xa + yp = 1. So?)
Now let ap be a nonzero element in Z/pZ. Show that a, is invertible. (Hint: Invoking the fact that a and p are relatively prime, we find that there must exist integers x and y such that xa + yp = 1. So?)
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
abstract algebra
![Example 2.58
However, Examples 2.55 and 2.56 do generalize suitably: it turns out
that for any prime p, the ring Z/pZ is a field (with p elements). Recall
from the discussions in Examples 2.20 and 2.21 that the elements of
Z/pZ are equivalence classes of integers under the relation a ~bif and
only if a-b is divisible by p. The equivalence class [a], of an integer a is
thus the set of integers of the form a + p, a ± 2p, a ±3p, .... Addition
and multiplication in Z/pZ are defined by the rules
1. [a]p + [b, = [a + blp
2. [a), (b), = [a - b)p](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3d940ce8-cba2-4a95-af25-aae0739ca5aa%2F25fb4e71-ac73-4aa7-8df6-c2994679c519%2Fh8prl8q_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Example 2.58
However, Examples 2.55 and 2.56 do generalize suitably: it turns out
that for any prime p, the ring Z/pZ is a field (with p elements). Recall
from the discussions in Examples 2.20 and 2.21 that the elements of
Z/pZ are equivalence classes of integers under the relation a ~bif and
only if a-b is divisible by p. The equivalence class [a], of an integer a is
thus the set of integers of the form a + p, a ± 2p, a ±3p, .... Addition
and multiplication in Z/pZ are defined by the rules
1. [a]p + [b, = [a + blp
2. [a), (b), = [a - b)p
Transcribed Image Text:Exercise 2.58.3
Now let [ap be a nonzero element in Z/pZ. Show that [a, is
invertible. (Hint: Invoking the fact that a and p are relatively
prime, we find that there must exist integers x and y such that
xa + yp = 1. So?)
Expert Solution
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 1 images
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
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…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
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…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
