Simple field extension where primitive element is algebraic

Question

I know that when $E$ is a field extension of field $F,$ $t \in E$\ $F,$ $t$ algebraic over $F,$ than the smallest field, containing $F \cup t$ is the same as set of polynomials, with coefficients in $F$ and a variable $t.$

Do you know how to prove it? Or maybe you've seen a proof of that statement?

Answer 1

Hint: If $E$ is an extension of $F$ and $t\in E$, then the ring extension $F[t]$ is also a field extension.