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Example 5: Give an explicit construction of a field with 4 elements. Solution: By Theorem 5, you know that there exists a finite field with 2² = 4 elements, viz. F₂2 = F4, with characteristic 2. You also know that F is the splitting field of x*-x over F₂. Let F₁ = {0,1,α,B). Now, x*- x = x(x − 1)(x²+x+1). By a direct verification, you can see that x² + x +1 is irreducible over F₂. Also, a, ß have to be the roots of x²+x+1. So a² +a+1=0. Also, relating the coefficients and the roots of x²+x+1₂a+B= -1 = 1, since char F₁ = 2.³ Request So B= a +1. explain Thus, F₁ = {0,1,0, 1+a), and a basis of F, over F₂ is {1,α). ***

Example 5: Give an explicit construction of a field with 4 elements. Solution: By Theorem 5, you know that there exists a finite field with 2² = 4 elements, viz. F₂2 = F4, with characteristic 2. You also know that F is the splitting field of x*-x over F₂. Let F₁ = {0,1,α,B). Now, x*- x = x(x − 1)(x²+x+1). By a direct verification, you can see that x² + x +1 is irreducible over F₂. Also, a, ß have to be the roots of x²+x+1. So a² +a+1=0. Also, relating the coefficients and the roots of x²+x+1₂a+B= -1 = 1, since char F₁ = 2.³ Request So B= a +1. explain Thus, F₁ = {0,1,0, 1+a), and a basis of F, over F₂ is {1,α). ***

Elements Of Modern Algebra
Elements Of Modern Algebra
8th Edition
ISBN:9781285463230
Author:Gilbert, Linda, Jimmie
Publisher:Gilbert, Linda, Jimmie
Chapter5: Rings, Integral Domains, And Fields
Section5.3: The Field Of Quotients Of An Integral Domain
Problem 13E
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Example 5: Give an explicit construction of a field with 4 elements. Solution: By Theorem 5, you know that there exists a finite field with 2² = 4 elements, viz. F₂ = F, with characteristic 2. You also know that F is the splitting field of x*-x over F₂. Let F₂ = {0,1,α,ß}. Now, x¹ - x = x(x − 1)(x² + x + 1). By a direct verification, you can see that x²+x+1 is irreducible over F₂. Also, a, ß have to be the roots of x²+x+1. So a² +a+1=0. Also, relating the coefficients and the roots of x² +x+1, α+B=-1=1, since char F, = 2. So B= a +1. Thus, F₁ = {0,1,0, 1+a), and a basis of F, over F₂ is (1,α). *** Request explain
Transcribed Image Text:Example 5: Give an explicit construction of a field with 4 elements. Solution: By Theorem 5, you know that there exists a finite field with 2² = 4 elements, viz. F₂ = F, with characteristic 2. You also know that F is the splitting field of x*-x over F₂. Let F₂ = {0,1,α,ß}. Now, x¹ - x = x(x − 1)(x² + x + 1). By a direct verification, you can see that x²+x+1 is irreducible over F₂. Also, a, ß have to be the roots of x²+x+1. So a² +a+1=0. Also, relating the coefficients and the roots of x² +x+1, α+B=-1=1, since char F, = 2. So B= a +1. Thus, F₁ = {0,1,0, 1+a), and a basis of F, over F₂ is (1,α). *** Request explain
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