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).
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).
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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![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).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Fc4fb56a5-afa6-4c2e-a167-f33c5b7cbfb2%2F1689cc0a-000e-4b13-872b-2f730289553e%2Fq8qe8j_processed.jpeg&w=3840&q=75)
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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