66. Let p be an odd prime and let S= {[1], [2],.... [p- 1]} be the set of nonzero equivalence classes mod p. a) Prove that for any integer a such that [a] e S, [-a] e S and [-a] #[a]. b) Prove that if [a] e S and [b] e S, then [a] © [b] e S. c) Prove that if [a]² = [b]? then [a] = [b] or [a] = [-b]. *d) Prove that exactly (p– 1)/2 members of S have square roots ([a] has a square root provided there is an integer x such that [x] = [a]).

Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
icon
Related questions
icon
Concept explainers
Question
??
66. Let p be an odd prime and let S = {[1], [2],...,[p- 1]} be the set of nonzero
equivalence classes mod p.
%3D
a) Prove that for any integer a such that [a] e S, [-a] e S and [-a] # [a].
b) Prove that if [a] e S and [b] e S, then [a] O [b] e S.
c) Prove that if [a]? = [b]? then [a] = [b] or [a] = [-b].
*d) Prove that exactly (p- 1)/2 members of S have square roots ([a] has a
square root provided there is an integer x such that [x]? = [a]).
Transcribed Image Text:66. Let p be an odd prime and let S = {[1], [2],...,[p- 1]} be the set of nonzero equivalence classes mod p. %3D a) Prove that for any integer a such that [a] e S, [-a] e S and [-a] # [a]. b) Prove that if [a] e S and [b] e S, then [a] O [b] e S. c) Prove that if [a]? = [b]? then [a] = [b] or [a] = [-b]. *d) Prove that exactly (p- 1)/2 members of S have square roots ([a] has a square root provided there is an integer x such that [x]? = [a]).
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 4 steps

Blurred answer
Knowledge Booster
Points, Lines and Planes
Learn more about
Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, advanced-math and related others by exploring similar questions and additional content below.
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
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…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,