Question about totally a transcendental extension

Question

Let $E/F$ be a totally transcendental extension, i.e., any element in $x\in E-F$ is transcendental over $F$. Is there any field $L$ so that for some $\alpha \in L-E$, $\alpha$ is algebraic over $F$?

In other words, can we say that if $E$ is a totally transcendental extension over $F$, then any field containing $E$ is also a totally transcendental extension over $F$?

Answer 1

Even if $E$ is totally transcendental over $F$, a field $L$ containing $E$ need not be totally transcendental over $F$. For example, let

$$F = \mathbb{Q},\;\;\;E=\mathbb{Q}(x),\;\;\;L=\mathbb{C}(x),\;\;\;\alpha=i$$

Answer 2

Take $F=Q, E=Q(X)$ and $L=E[√2]$