Are $x^2+1$ and $x+1$ the same in $\mathbb{Z}_2[x]$?

Question

Are $x^2+1$ and $x+1$ the same in $\mathbb{Z}_2[x]$?

I can imagine two answers for this (and hopefully someone can tell me which one, if any, is the right one):

• Yes, because under the “Evaluation homomorphism” $x^2+1$ and $x+1$ are the same.

That is, evaluating $x^2+1$ at $0$ and $1$ gives the same results as evaluating $x+1$ at $0$ and $1$.

• No, $x^2+1$ and $x+1$ are different (formal expressions).

This means we should treat the polynomials as formal expressions and ignore the fact that under evaluation they are the same. Evaluation is irrelevant.

Any help?

Answer 1

There are two different things:

• for any $x\in \mathbb{Z}/2\mathbb{Z}$, $x^2+1=x+1$;

• the polynomials $x^2+1$ and $x+1\in \mathbb{Z}/2\mathbb{Z}[x]$ are different.

Depending on what meaning you give to "$x^2+1=x+1$ in $\mathbb{Z}/2\mathbb{Z}$", one of these answers apply.

Answer 2

What you have written is correct. Viewed as functions $\mathbb{Z}/2\mathbb{Z}\to \mathbb{Z}/2\mathbb{Z}$, $x^2+1$ and $x+1$ are identical. As elements of the ring $(\mathbb{Z}/2\mathbb{Z})[x]$ they are necessarily distinct. Indeed, one is degree $2$ while the other is degree $1$.