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(4) Express each member of F(u) in terms of u.

(4) Express each member of F(u) in terms of u.

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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please answer number 4
Let F = Z₂ and let f(x) = x³ + x + 1 € F[x]. Suppose thataisa zero of f(x) in some extension of F. How many elements does F(a) have? Express each member of F(a) in terms of a. Write out a com- plete multiplication table for F(a). (1) Show that f is irreducible over F. Use the irreducibility theorem about degree 3 polynomials over F. (2) Construct an extension field F(u) as in our examples such that f has a root u. (3) How many elements does F(u) have? Why? (4) Express each member of F(u) in terms of u. (5) Write out a complete multiplication table for F(u).
Transcribed Image Text:Let F = Z₂ and let f(x) = x³ + x + 1 € F[x]. Suppose thataisa zero of f(x) in some extension of F. How many elements does F(a) have? Express each member of F(a) in terms of a. Write out a com- plete multiplication table for F(a). (1) Show that f is irreducible over F. Use the irreducibility theorem about degree 3 polynomials over F. (2) Construct an extension field F(u) as in our examples such that f has a root u. (3) How many elements does F(u) have? Why? (4) Express each member of F(u) in terms of u. (5) Write out a complete multiplication table for F(u).
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