1

I am trying to figure out how to prove that $\mathbb Z_2[x]/(x^3+x^2+1)$ and $\mathbb Z_2[x]/(x^3+x+1)$ are isomorphic.

I know these sets all have the form $\{a+bx+cx^2\mid a,b,c\in\mathbb Z_2\}$. Is there any idea about it?

user26857
  • 52,094
functions
  • 181
  • You could find an explicit isomorphism from one to the other. Or, if you have developed some of the theory of finite fields, you could prove that they have the same number of elements, and then cite a theorem that says that, up to isomorphism, there is at most one field with any given finite number of elements. So, it really depends on what tools you have available. – Gerry Myerson Apr 08 '19 at 13:23

2 Answers2

5

$X\mapsto X+1$ is an automorphism of $\mathbb Z_2[X]$ which sends $X^3+X+1$ to $X^3+X^2+1$.

J. De Ro
  • 21,438
user26857
  • 52,094
2

Since both the polynomials $(x^3+x+1)$ and $(x^3+x^2+1)$ are irreducible in $\mathbb{Z}_2[x]$, both $ \mathbb{Z}_2[x]/(x^3+x+1)$ and $\mathbb{Z}_2[x]/(x^3+x^2+1)$ are fields of order $8$ and are isomorphic.

MAS
  • 10,638