Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
1. The goal of this problem is to determine all of the roots of the polynomial f = X²+3X+1 in Z/209Z. Note that 209 = 11. 19. (a) Verify that the discriminant of f is equal to 5. (b) Determine the square roots of 5 + 11Z. (c) Determine the square roots of 5 + 19Z.

1. The goal of this problem is to determine all of the roots of the polynomial f = X²+3X+1 in Z/209Z. Note that 209 = 11. 19. (a) Verify that the discriminant of f is equal to 5. (b) Determine the square roots of 5 + 11Z. (c) Determine the square roots of 5 + 19Z.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
See similar textbooks
icon
Related questions
Question
1. The goal of this problem is to determine all of the roots of the polynomial f = X²+3X+1 in Z/209Z. Note that 209 = 11. 19. (a) Verify that the discriminant of f is equal to 5. (b) Determine the square roots of 5 + 11Z. (c) Determine the square roots of 5 + 19Z. (d) Use the quadratic formula to compute the roots of f in F11. (e) Use the quadratic formula to compute the roots of f in F19. (f) Write out the four systems of congruences which give (via their solutions) the roots of f in Z/209Z. (g) Compute (11 + 19Z)-¹. (This will be helpful in the next part of the problem.) (h) By solving the systems you gave in (f), determine the roots of f in Z/209Z. Express each root using its least nonnegative representative.
Transcribed Image Text:1. The goal of this problem is to determine all of the roots of the polynomial f = X²+3X+1 in Z/209Z. Note that 209 = 11. 19. (a) Verify that the discriminant of f is equal to 5. (b) Determine the square roots of 5 + 11Z. (c) Determine the square roots of 5 + 19Z. (d) Use the quadratic formula to compute the roots of f in F11. (e) Use the quadratic formula to compute the roots of f in F19. (f) Write out the four systems of congruences which give (via their solutions) the roots of f in Z/209Z. (g) Compute (11 + 19Z)-¹. (This will be helpful in the next part of the problem.) (h) By solving the systems you gave in (f), determine the roots of f in Z/209Z. Express each root using its least nonnegative representative.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 5 steps with 3 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
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,
SEE MORE TEXTBOOKS