We're trying to understand the field Z2[x]⟨x3+x2+1⟩, and in particular why it has eight elements. I know that x3+x2+1 is irreducible in Z2 and so factor is going to be a field.The issue I'm having is that the notes claim that (x+I) has an inverse in the field, where I=⟨x3+x2+1⟩. They write(x+I)(x2+x+I)=x3+x2+I=1+II don't understand the last line, I know that multiplication of cosets is defined that way, I just don't see why we get 1 at the end. I think it's possible I didn't fully understand cosets and ideals but I can't fully articulate what the issue is. Any help towards filling in the gaps of my understanding would be greatly appreciated.

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We&#039;re trying to understand the field Z2[x]⟨x3+x2+1⟩, and in particular why it has eight elements. I know that x3+x2+1 is irreducible in Z2 and so factor is going to be a field.The issue I&#039;m having is that the notes claim that (x+I) has an inverse in the field, where I=⟨x3+x2+1⟩. They write(x+I)(x2+x+I)=x3+x2+I=1+II don&#039;t understand the last line, I know that multiplication of cosets is defined that way, I just don&#039;t see why we get 1 at the end. I think it&#039;s possible I didn&#039;t fully understand cosets and ideals but I can&#039;t fully articulate what the issue is. Any help towards filling in the gaps of my understanding would be greatly appreciated.

levurdondishav4 · Accepted Answer

Step 1 The might know that a+I=b+I⇔a−b∈I So (x3+x2)−1=(x3+x2+1)∈I So (x3+x2)+I=1+I also any element in Z2[x](x3+x2+1) is of the form (ax2+bx+c)+I. There are 2 choices for each a, b, c. So there are 8 distinct elements in Z2[x] by a cubic polynomial, you get a remainder which is a quadratic polynomial of the form ax2+bx+c So if p(x)∈Z2[x]] then p(x)=q(x)(x3+x2+1)+r(x) where r(x)=ax2+bx+c. For some a, b, c∈Z2 So p(x)+I=ax2+bx+c+I as q(x)(x3+x2+1)∈⟨x3+x2+1⟩

Thomas Lynn · Accepted Answer

This is because   x3+x2+1∈I  by definition and therefore   (x3+x2+1)+I=I  meaning that,  (x3+x2)+I=(x3+x2)+(x3+x2+1)+I=1+I  because the characteristic is 2.

We're trying to understand the field \frac{\mathbb{Z}_{2}

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