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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
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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.
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