Let $K \subseteq L \subseteq M$ be two field extensions then, $K \subseteq M$ is algebraic iff $K\subseteq L$ and $L\subseteq M$ are both algebraic.

Question

Let $K\subseteq L\subseteq M$ be two field extensions then, $K\subseteq M$ is algebraic, if and only if, $K\subseteq L$ and $L\subseteq M$ are both algebraic.

Please how can I prove the sufficient condition

Answer 1

For one direction, suppose $K\subseteq M$ is an algebraic field extension. Then every element of $M$ satisfies a polynomial with coefficients in $K$; in particular, every element of $L\subseteq M$ does too, and so $K\subseteq L$ is an algebraic extension. Additionally, polynomials over $K$ can be considered as polynomials over $L$ via the inclusion $K\subseteq L$, so every element of $M$ hence satisfies a polynomial over $L$ and thus $L\subseteq M$ is an algebraic extension.

For the reverse direction, the main point is that, if $E\subseteq F$ is a field extension, then $f_1,\dots,f_n\in F$ are each algebraic over $E$ if and only if $[E(f_1,\dots,f_n):E]<\infty$. This can be proved by induction on $n$ and the tower law; I will leave it as an exercise for you.

Anyway, this fact makes the desired result easy to show. Suppose $K\subseteq L$ and $L\subseteq M$ are algebraic, and let $m\in M$. Then $m$ is algebraic over $L$, so there is a polynomial with coefficients in $L$ satisfied by $m$. Let $l_1,\dots,l_n$ be these coefficients. Since $L$ is algebraic over $K$, each $l_i$ is algebraic over $K$, so $[K(l_1,\dots,l_n):K]<\infty$ by the result mentioned above. Also, $m$ is algebraic over $K(l_1,\dots,l_n)$, and so, again by the result mentioned above, $$[K(l_1,\dots,l_n,m):K(l_1,\dots,l_n)]<\infty.$$ By the tower law, this means $[K(l_1,\dots,l_n,m):K]<\infty$, and so, by one final use of the result mentioned above, $m$ is algebraic over $K$.