Example of an infinite algebraic field extension which is not simple

Question

I want to an example of an infinite algebraic field extension that is not simple. I was thinking of the algebraic numbers $\mathbb{A}$ over $\mathbb{Q}$, or all the square roots of primes adjoined to $\mathbb{Q}$ as an example, but I wasn't sure how to show it is never equal to $\mathbb{Q}(\theta)$ for any $\theta$.

Answer 1

Simple algebraic extensions are finite. Since $[\overline{\Bbb Q}:\Bbb Q]=\infty$, we are done. The extension $\overline{\Bbb Q}\mid \Bbb Q$ cannot be simple.

References:

An algebraic simple extension has finite degree

Algebraic closure of $\mathbb{Q}$ in $\mathbb{C}$. Alternative proof?