I am trying to solve a problem in Chapter 15 about finite fields in Artin's book. It gives me $f(x)=x^3+x+1$ and $g(x)=x^3+x^2+1$ which are irreducible over $\mathbb{F_2}$. $K$ and $L$ are field extensions of $\mathbb{F_2}$ by adjoing a root of $f$ and $g$ respectively . I was asked to describe the isomorphism from $K$ to$L$ and determine the number of such isomorphisms.
Assume $\alpha$ and $\beta$ are the root of $f$ and $g$ and just map $\alpha \to \beta$ we can get the iso.
To determine the number of such isomorphism , I think that the map could only be of the form $\sigma:\alpha \to a\beta^s+b$ where $a,b \in \mathbb{F_2}$ and $s\in \mathbb{N}$. If $\sigma$ is an isomorphism, $a$ should be $1$ and b can be $1$ or $0$. I want to prove that $s$ can only be $1$ so that the number of such iso is 2. But I was stuck, and I am not sure if the number of such isomorphism is 2.
Can anyone give me some help? Thanks in advance!