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Let $\mathbb{Z}_2 [x]$ be the ring of all polynomials with coefficients in $\mathbb{Z}_2$. List the elements of the field $\mathbb{Z}_2 [x]/〈x^2+x+1〉$, and make an addition and multiplication table for the field.

I'm don't really understand what this notation means,$\mathbb{Z}_2 [x]/〈x^2+x+1〉$. Any help is appreciated.

rschwieb
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Cody S
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  • It's a quotient ring: see here or here. – Viktor Vaughn Oct 02 '17 at 05:18
  • So its a quotient ring. Ok, so is this a correct equivalency? $\mathbb{Z}_2[x]/\left \langle x^2+x+1 \right \rangle=\left { ax+b+\left \langle x^2+x+1 \right \rangle \right }$ Where do I go from here? – Cody S Oct 02 '17 at 05:22
  • How many cosets can you possibly have? – David Wheeler Oct 02 '17 at 05:27
  • In this case would the number of cosets be elements on the table or elements of $\mathbb{Z}_2[x]/\left \langle x^2+x+1 \right \rangle $ – Cody S Oct 02 '17 at 05:29
  • Each coset contains a representative that is remainder of a polynomial after dividing $x^2+x+1$. How many possible remainders are there? Note that the remainder must be of smaller degree than $x^2+x+1$. –  Oct 02 '17 at 05:31
  • $4$ or so? $0,1,x,x+1$? – Cody S Oct 02 '17 at 05:34
  • Yup so now you can construct your tables using these representatives :D –  Oct 02 '17 at 05:36
  • And if a polynomial has degree $\ge 2$, by polynomial division, it is a representative of one of the $4$ cosets that we already know. –  Oct 02 '17 at 05:42
  • @CodyS Next time, use a more useful title. People are more likely to look at a title that gives them some idea of what they are about to answer. Nobody cares to look at titles like "random abstract algebra question help!" I also picked some more useful tags for you. – rschwieb Oct 02 '17 at 13:10

1 Answers1

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To try to cover all the bases:

$\mathbb{Z}_2 [x]$ is the polynomial ring with coefficients in $\mathbb Z_2$, the field of two elements.

$\langle x^2+x+1\rangle$ is the ideal generated by the polynomial $x^2+x+1$ in $\mathbb{Z}_2 [x]$.

$\mathbb{Z}_2 [x]/\langle x^2+x+1\rangle$ is the quotient ring with respect to this ideal. That is, it is the set of cosets of the equivalence relation $x\sim y \iff x-y\in \langle x^2+x+1\rangle$ with the natural ring structure.

Intuitively, in this quotient every element of $\mathbb Z_2[x]$ is equivalent to some polynomial of degree $1$ or less.

rschwieb
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