I'm working on a question saying $A \subseteq B \subseteq C$ fields, and $a \in C$ algebraic over $A$, either prove $[A(a):A] ≥ [B(a):B]$ or give a counter example.
I think it's true and used the following equation: $[B(a):A(a)][A(a):A] = [B(a):B][B:A]$, and I think we can show that $[B(a):A(a)] < [B:A]$ by using the classic definition regarding basis of field extensions, but I'm not sure how to do so since I don't even know if it's a finite extension.