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Galois's Construction of his Fields

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Let beta be the Galois imaginary associated with the irreducible polynomial
x^3 + x^2 + 1 over mod2. Solve the (systems of simultaneous) equations in GF(2, x^3+x^2+1).

1) (1+ beta)x + beta = 1 + beta^2

2) x + y = beta and x + beta*y = 1

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Galois's Construction of his Fields is clarified in this solution.

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Galois's Construction of his Fields

Let beta be the Galois imaginary associated with the irreducible polynomial
x^3 + x^2 + 1 over mod2. Solve the (systems of simultaneous) equations in GF(2, x^3+x^2+1).

Solution:

1) (1+ beta)x + beta = 1 + beta^2

2) x + y = beta and x + beta*y = 1

1) We have beta^3+beta^2+1=0. So ...

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