This is from an example in my textbook:(adsbygoogle = window.adsbygoogle || []).push({});

"The polynomial x^2+x+1 has no root in Z mod 2, and so is irreducible. Hence the required field is:

F = Z_2[x] / <x^2 + x + 1> = {a + bt | a,b are elements of Z_2; t^2+t+1 = 0}.

Thus F = {0, 1, t, 1+t} and t^2 = t + 1."

I have two questions.

My first question is if this a exercise problem rather than an example, I would have had no idea to grab the polynomial x^2+x+1 in Z mod 2. Can somebody explain why this was apparently so obvious?

My next question is why is t^2 = t + 1? Wouldn't t^2 = -t - 1?

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# Constructing a field of 4 elements

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