suppose K is an extension field of finite degree, and L,H are middle fields such that L(H)＝K.Prove that [K:L]≤[H:F]

Question

Suppose $K$ is an extension field of finite degree, and $L,H$ are middle fields such that $L(H)＝K$.Prove that $[K:L]≤[H:F]$

Well, I just have no idea about how to use the condition that $L(H)＝K$. Please give me some help, thanks!

Answer 1

Let $F$ be the base field. Since $[K:F]<\infty$, then $K$ is algebraic over $F$. Similarly by the tower theorem, $H$ is algebraic over $F$ and $L$ is algebraic over $F$. This means that $H=F(\beta_{1},\dots,\beta_{m})$ for some $\beta_{1},\dots,\beta_{m}$ algebraic over $F$, so that $[H:F]=m$, and $L=F(\alpha_{1},\dots,\alpha_{n})$ for some $\alpha_{1},\dots,\alpha_{n}$ algebraic over $F$, so that $[L:F]=n$. Since $L(H)=K$, then \begin{equation*} K=L(\beta_{1},\dots,\beta_{m})=F\left(\alpha_{1},\dots,\alpha_{n},\beta_{1},\dots,\beta_{m}\right) \end{equation*} so that $[K:F]\leq m+n$, and $[K:L]\leq m=[H:F]$ (draw a diagram to help you visualise this).