An isomorphism between $\mathbb F_2(\alpha )$ and $\mathbb F_2(\beta )$.

Question

We know that $f(x)=x^3+x+1$ and $g(x)=x^3+x^2+1$ are irreducible over $\mathbb F_2$. Let $\alpha $ a root of $f$ and $\beta $ a root of $g$. Let $K=\mathbb F_2(\alpha )$ and $L=\mathbb F_2(\beta )$. Find explicitely an isomorphism between $K$ and $L$.

Attempts

Does $\alpha \longmapsto \beta $ work ? It look to easy. If not, how can I do ?

Answer 1

Not quite. You have to find all roots of the polynomials over $\mathbb{F}_2(\alpha)$. In fact, for $x^3+x+1$ all roots are given by $\alpha$, $\alpha^2,\alpha^2+\alpha$.

Answer 2

Hint: check that $\beta+1$ satisfies the polynomial $f(x)$ for $\alpha$.

Answer 3

You have to map $\alpha$ to a root of the polynomial $x^3 + x + 1$ in $\Bbb F(\beta)$.

You can just try them all: $\Bbb F(\beta)$ has $8$ elements, $0, 1, \beta$ are not a root, so there are $5$ candidates left and $3$ of them are a root.