Algebra's exercise about algebraic extension of fields

Question

Could you help me with this exercise?

Let $\alpha \in L:K$ with $L$ algebraic over $K$ and let $$\begin{pmatrix} a & b \\ c & d \end{pmatrix} \in GL_2(K).$$ Let $\beta=\frac{a \alpha+b}{c\alpha+d}$. Prove that $K(\beta)=K(\alpha)$ and find an expression for $m_{\beta}(x)$.

I found that $\alpha=\frac{b-\beta d}{c\beta-a}$ and I tried to use it to understand how $m_{\beta}$ is made but I didn't manage to.

Thanks for the support!

Answer 1

• $K(\beta) \subseteq K(\alpha)$ because $\beta \in K(\alpha)$

• $K(\alpha) \subseteq K(\beta)$ because $\alpha \in K(\beta)$ since that matrix is invertible