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5. Consider the polynomial f = X² + X + 8. (a) Determine the roots of f in F5 using brute force. Express the roots using least nonnegative integer representatives. (b) Determine the roots of f in F7 using brute force. Express the roots using least nonnegative integer representatives. (c) Write out the four systems of congruences which give (via their solutions) the roots of f in Z/35Z. (d) By solving the systems you gave in (c), determine the roots of f in Z/35Z. Express the roots using least nonnegative integer representatives.

5. Consider the polynomial f = X² + X + 8. (a) Determine the roots of f in F5 using brute force. Express the roots using least nonnegative integer representatives. (b) Determine the roots of f in F7 using brute force. Express the roots using least nonnegative integer representatives. (c) Write out the four systems of congruences which give (via their solutions) the roots of f in Z/35Z. (d) By solving the systems you gave in (c), determine the roots of f in Z/35Z. Express the roots using least nonnegative integer representatives.

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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5. Consider the polynomial f = X² + X + 8. (a) Determine the roots of f in F5 using brute force. Express the roots using least nonnegative integer representatives. (b) Determine the roots of f in F7 using brute force. Express the roots using least nonnegative integer representatives. (c) Write out the four systems of congruences which give (via their solutions) the roots of f in Z/35Z. (d) By solving the systems you gave in (c), determine the roots of f in Z/35Z. Express the roots using least nonnegative integer representatives.
Transcribed Image Text:5. Consider the polynomial f = X² + X + 8. (a) Determine the roots of f in F5 using brute force. Express the roots using least nonnegative integer representatives. (b) Determine the roots of f in F7 using brute force. Express the roots using least nonnegative integer representatives. (c) Write out the four systems of congruences which give (via their solutions) the roots of f in Z/35Z. (d) By solving the systems you gave in (c), determine the roots of f in Z/35Z. Express the roots using least nonnegative integer representatives.
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