19.3.5. Let p(x) = x³ − x + 1 Є F2[x]. Let I = (p(x)), and let R = F2[x]/I. Let a=I+xЄ R. (a) Is p(x) irreducible in F2[x]? (b) How many elements does R have? (c) Is a a unit in R? (d) What is the additive order of a? What is the multiplicative order of a? (e) What is the characteristic of R?
19.3.5. Let p(x) = x³ − x + 1 Є F2[x]. Let I = (p(x)), and let R = F2[x]/I. Let a=I+xЄ R. (a) Is p(x) irreducible in F2[x]? (b) How many elements does R have? (c) Is a a unit in R? (d) What is the additive order of a? What is the multiplicative order of a? (e) What is the characteristic of R?
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.3: Zeros Of Polynomials
Problem 32E
Question
![19.3.5. Let p(x) = x³ − x + 1 Є F2[x]. Let I = (p(x)), and let R = F2[x]/I. Let
a=I+xЄ R.
(a) Is p(x) irreducible in F2[x]?
(b) How many elements does R have?
(c) Is a a unit in R?
(d) What is the additive order of a? What is the multiplicative order of
a?
(e) What is the characteristic of R?](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff5fbaae5-8d47-4476-8095-8b380294ae7e%2F468a99d6-d921-4dd9-9243-9364d2bff611%2F4uusqap_processed.png&w=3840&q=75)
Transcribed Image Text:19.3.5. Let p(x) = x³ − x + 1 Є F2[x]. Let I = (p(x)), and let R = F2[x]/I. Let
a=I+xЄ R.
(a) Is p(x) irreducible in F2[x]?
(b) How many elements does R have?
(c) Is a a unit in R?
(d) What is the additive order of a? What is the multiplicative order of
a?
(e) What is the characteristic of R?
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