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The field is F,, the splitting field of x²-x, over F₂. Now x' - x = x(x - 1)(x² + x +...+x+1) = x(x − 1)(x-2)(x² + 1)(x² − x − 1)(x² + x - 1) as a product of irreducible polynomials over Z₁. how do we this If a is a root of x² +1 over Z₁, then {1, a) is an F-basis of F,.quest

The field is F,, the splitting field of x²-x, over F₂. Now x' - x = x(x - 1)(x² + x +...+x+1) = x(x − 1)(x-2)(x² + 1)(x² − x − 1)(x² + x - 1) as a product of irreducible polynomials over Z₁. how do we this If a is a root of x² +1 over Z₁, then {1, a) is an F-basis of F,.quest

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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i) ii) i) ii) Construct a field with 9 elements. Determine the irreducible polynomial satisfied by each element of the field in (i). The field is F,, the splitting field of x2-x, over F₂. Now x' - x = x(x-1)(x² + x +...+x+1) = x(x − 1)(x-2)(x² +1)(x² − x − 1)(x²+x-1) as a product of irreducible polynomials over Z,. how do we this get If œ is a root of x² +1 over Z3, then {1, α) is an F-basis of F₂. lequest So F, = {0, 1, 2, α, 2a, 1+ α, 1+2a, 2+α, 2+2α). expan this step 0, 1, 2 satisfy x,x-1, x-2, respectively. a, 2a satisfy x² +1, 1+ α, 1+ 2a satisfy x²+x-1 2+α, 2+2a satisfy x²-x-1.
Transcribed Image Text:i) ii) i) ii) Construct a field with 9 elements. Determine the irreducible polynomial satisfied by each element of the field in (i). The field is F,, the splitting field of x2-x, over F₂. Now x' - x = x(x-1)(x² + x +...+x+1) = x(x − 1)(x-2)(x² +1)(x² − x − 1)(x²+x-1) as a product of irreducible polynomials over Z,. how do we this get If œ is a root of x² +1 over Z3, then {1, α) is an F-basis of F₂. lequest So F, = {0, 1, 2, α, 2a, 1+ α, 1+2a, 2+α, 2+2α). expan this step 0, 1, 2 satisfy x,x-1, x-2, respectively. a, 2a satisfy x² +1, 1+ α, 1+ 2a satisfy x²+x-1 2+α, 2+2a satisfy x²-x-1.
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