$\mathbb{Z}_2[x]$ denotes the set of all polynomials with coefficients that are $0$ or $1$.
My book hints that there are only $4$ elements in this factor ring, and each nonzero one has an inverse. (Side note, multiplication and addition of coefficients is done modulo $2$ in this factor ring).
My question:
Why are there only $4$ elements in this group? It seems to be there would be infinitely many, since it seems $x^n$is in $\mathbb{Z}_2[x]$, and $x^n + \langle x^2+x+1 \rangle \neq x^m + \langle x^2+x+1\rangle$ if $n\neq m$; why is that incorrect?
Once I understand this I think I will be able to write a proof that shows every element has an inverse, but my confusion on what elements this ring actually contains is preventing me from attempting that now.
Best Answer
The only polynomials that could be in the field would be $0, 1, x, x+1, x^2, x^2+1, x^2+x$. However, $$x^2+x = x^2+x - (x^2+x+1) = -1 = 1$$ $$x^2+1 = x^2+1 - (x^2+x+1) = -x = x$$ $$x^2 = x^2 - (x^2+x+1) = -(x+1) = x+1$$ so the only elements are $0, 1, x, x+1$.