Prove that $\Bbb Z_2(\alpha)=\Bbb Z_2(\beta)$ .

Question

Consider the polynomials $x^3+x^2+1,x^3+x+1$ over $\Bbb Z_2$ which have roots say $\alpha,\beta $ respectively. Prove that $\Bbb Z_2(\alpha)=\Bbb Z_2(\beta)$ .

Since both $x^3+x^2+1,x^3+x+1$ are irreducible so $\Bbb Z_2(\alpha)\cong \Bbb Z_2[x]/<x^3+x^2+1>$ and $\Bbb Z_2(\beta)\cong \Bbb Z_2[x]/<x^3+x+1>$.

The elements of $\Bbb Z_2(\alpha)$ are $\{a+b\alpha +c\alpha^2 \mid a,b,c\in \Bbb Z_2\}$. Also, the elements of $\Bbb Z_2(\beta)$ are $\{a+b\beta+c\beta^2 \mid a,b,c\in \Bbb Z_2\}$.

How to show they are equal?

Answer 1

The polynomial of $1/\alpha$ is $x^3+x+1$, so $$\mathbb{Z}(\alpha)=\mathbb{Z}(1/\alpha)\equiv\mathbb{Z}(\beta)$$