I am doing an intro research project on closure properties of algebraic numbers. I don't have a strong grasp of Galois theory. I wanted to understand the implication of this question and whether it was worth exploring
Can a a collection of mathematical objects which is not algebraic over a field be algebraic over a field extension? if it is worth exploring can I maybe get an example of when this is possible?
$\pi$ is not algebraic over $\Bbb Q$. But after extending $\Bbb Q$ to $\Bbb Q(\pi)$, suddenly $\pi$ becomes algebraic (it's even contained in the field itself). As a second example, $\sqrt \pi$ is an element which becomes algebraic without becoming an element of the field.
However, if $K/\Bbb Q$ is an algebraic field extension then there is no way $\pi$ can be algebraic over $K$. That's because if $f(\pi) = 0$ for some polynomial with coefficients in $K$, one can use the fact that $K$ is algebraic to find a polynomial $g(x)$ with coefficients in $\Bbb Q$ such that $g(\pi) = 0$.
So, performing an algebraic field extension cannot change which elements are algebraic and which are transcendental, but a transcendental field extension can.