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. f (2) Construct an extension field F(u) as in our examples such that ƒ has a root u. (3) How many elements does F(u) have? Why?
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. f (2) Construct an extension field F(u) as in our examples such that ƒ has a root u. (3) How many elements does F(u) have? Why?
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.1: Polynomial Functions Of Degree Greater Than
Problem 36E
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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%2F2ddc7a09-9ee5-460c-abf4-dd58e05c623c%2Ff3a03b48-425b-4b65-8b86-000eec17750c%2Fibhirmn_processed.png&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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