Say $K/F$ is a field extension. The elements in $K$ that are algebraic over $F$ form a subfield of $K$. Is this subfield isomorphic to $F[t]$? What would this isomorphism look like?
This is not a textbook problem. This is what I inferred from a statement made in a well-known textbook on Algebraic Geometry. I only want to verify if what I inferred is true.
Notes:
$F[t]$ is a field iff $t$ is algebraic over $F$. When $t$ is algebraic over $F$, then you can construct this field as the quotient $F[t]\cong F[X]/(m)$ where $m$ is the minimal polynomial of $t$ over $F$. The mapping is the obvious one.
It is not common for a subfield of algebraic elements to be just generated by a single $t$. Consider $\Bbb Q$ and the elements algebraic over it inside $\Bbb C$.